I have always been under the impression that "may" was the more polite word to use when ordering food. At a fast food place I suppose it isn't too big of a deal, but if I were taking an order I would probably appreciate someone using may over can.

It is interesting that you pointed out that you could be asking if an item is available for you to order, which seems to be a perfectly valid interpretation as well. Without thinking about it too much I would generally just go with ‘May I...’.

In the dilemma "may" vs. "can" and which form is preferable, it depends on how old the speaker is, where they live and which dialect of English they speak.

There is an age-old debate that can in requests, is asking if something is "possible", e.g.

A: Can I have a glass of water?
B: Yes, you can (=it is possible).

Whereas may, some argue, is asking permission and is used in polite, slightly hesitant, requests.

A: May I have a glass of water?
B: Yes, (you may) I'll pour you a glass.

Older, and perhaps more educated, British speakers will probably prefer using "may", and perhaps most Americans too, regardless of their level of education. In my experience, younger British speakers will nearly always ask "Can I...?"

The modal could is also used in formal, rather polite, requests

Could I have a glass of water, please?

There is a nice summary by Cambridge Dictionary titled Requests: Asking for something And in this six-year-old question posted on EL&U: Difference between "can" and "may"

EDIT (June 5 2018)

No waiter that I know of, would think to reply a customer's order with "Yes, you can/may have a glass of water" that could sound arrogant. A normal response would be “Yes, certainly”. The whole point of can/could/may/might/'d like (etc.) in requests is the acknowledgment that the server or counter attendant is the person who brings and sometimes even prepares the food or drink in person.

The server is doing their job but they're doing it (and many do) to the best of their ability. When a customer uses polite language, it is not sham courtesy but a sign of respect.

A server or waiting staff takes on a very important role in a restaurant which is to always be attentive and accommodating to the customers. Wikipedia

Yes, "can" is technically more correct - at least, if you believe the main reason for possible rejection would be unavailability of the item. And technically correct speech is often less polite.

With experience, you'll know that restaurants sometimes pad their menus with items available only occasionally, in some cases never. That saves time and money from reprinting, makes the menu look more attractive than the actually available narrow selection, and in some cases opens door to bait and switch strategies. By using "can", you are implying that this restaurant might be following these morally questionable practices and therefore shows distrust.

By using "may", you are only implying that sometimes ordered items cannot be had, and you have no clue why that might happen. Perhaps because it's meant for common folks and doesn't taste as awesome as you deserve. So this wording implies trust, and therefore, is more polite.

My personal preference for placing order is "I would have ..., thank you" which implies I completely trust them to deliver on their offering and does not resemble an awkward request for permission.

As a simple distinction:

Can

Can I have the apple pie?
Am I able to have the apple pie?

This is asking for a possibility.

• Yes.
• No, we don't have any apple pie left.

May

May I have the apple pie?
Am I allowed to have the apple pie?

This is asking for permission.

• Yes.
• No, because you didn't eat your vegetables.

In other words, it might be possible to give you apple pie, but you're not going to get any.

Politeness

That being said, they are commonly understood to mean the same thing. "May" sounds a bit old fashioned compared to "can", in cases where either is being used as a matter of politeness.
The distinction listed above is correct, but often obvious (or not relevant) in the current context.

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Note that this type of polite phrasing is often ambiguous:

Can you give me the money?

Can be understood in different ways:

• Are you allowed to give me the money?
• Are you able to give me the money?
• Please hand me the money.

The correct interpretation is often a matter of context. It can depend on who you ask, for example:

Can I have the apple pie?

• If you ask your waiter, then you're asking if it's possible to have apple pie. A waiter doesn't give permission.
• If you ask your mother (while at a restaurant), you're asking for her permission. She can't know if the kitchen has any apple pie.
• If you ask your mother (at home), it usually combines the questions, asking if you're both able and allowed to have it.
• If you ask someone who is closer to a distant plate of apple pie, it stand to reason that you're asking them to pass it to you.

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Some responses to what you said:

'May I' implies that what you're asking for is already known to be present and available.

It doesn't. You might assume that someone only asks this question if they already know the thing they're asking for is available, but that is your assumption.

As a very simple example, when I was a kid I would ask if I could order dessert before I looked at the dessert menu. My cousin would look at the menu, and ask if they were allowed dessert if they found something they wanted.

The assumption logic works both ways. I saw no reason to look at the menu (and get my hopes up) until I knew I was going to be allowed to have something. My cousin, on the other hand, only wanted to ask permission if it was going to be worth it (since they'd otherwise waste an opportunity).

'Can I' implies more unsurety

Both options imply uncertainty, but they express different uncertainties (possibility/permission).

Beyond Grammar and Into the Social Codes

If you have ever taught English, you might know that "May I have x" is a polite form. This really is not about grammar. It's about how certain expressions are used in certain situations and also functions as markers of a social code.

This is not about being technically correct, it's about being the most polite (one can be.) There are many variations on how to ask to be served something in a restaurant, but as far as may/can goes, the utterance marked as most polite is:

1) "May I have [dish or food] please?"

Here, "May I have [dish or food]" means: I am ordering this dish or food.

That said, one often hears "Can I have x" when ordering food, and it's acceptable and fine in contemporary American English since everybody uses it but will not pass an advanced politeness test.

Permission: May I x is also used to ask for actual permission to do something:

• May I leave my car in front of your house for the weekend?

You will hear, however, people asking for permission and saying: - Can I leave my car in front of your house.

And,naturally, this use of "Can I" for permission has zero to do with having the ability to do something as is: "Can he ride a horse or will I have to teach him how?"

In the US, you will hear guys (yes, mostly guys) go into a pizza shop and say: "Can I get (pronouncing it "git") a slice of pizza?" That's fine in a pizza shop. It is typically working class and you probably wouldn't hear "Can I git" in a five-star restaurant. Here's the nasty truth: There is a class issue here. People who say "May I" in a restaurant are immediately recognizable by others who would say that. There are all sorts of educational and social status implications in it. No one like to admit this but it is true.

2) "Shall I [do x for someone]." Shall I open the window?

Shall I [do x for someone] means: Do you want [person or pronoun] to do [something] for [someone].

"Shall I" is used to offer to do something for someone or to suggest doing something for someone.

My answer is to this question is not about grammar, but about politeness as strategy.

Politeness Strategies in English Grammar by Richard Nordquist Updated December 29, 2017 In sociolinguistics and conversation analysis (CA), politeness strategies are speech acts that express concern for others and minimize threats to self-esteem ("face") in particular social contexts.

Politeness as strategy, not grammar

Neither is technically correct.

A decent restaurant will inform you if certain menu items are unavailable for some reason before you order. Some places aren't good about this, but generally the assumption is if something's on the menu, it's available for purchase. I doubt your intention is to ask if it's possible to have a specific meal, so using "can" doesn't make much sense. To convince yourself of this, replace it with "Is it possible for me to have [dish]?" Sounds silly, doesn't it?

Another general assumption is that you are allowed to order anything on the menu. That's the whole purpose of a restaurant. Therefore, "may" doesn't make much sense either. Again, to convince yourself of this, replace it with "Am I allowed to have [dish]?" Of course that's an absurd question.

There are cases where it makes sense to say "can" or "may", such as if you're asking to make a substitution or get something extra added. In those situations, I'd say the two phrases are equivalent, since whether you are able to have your wish will depend entirely on whether they're willing to do it or not. There might be exceptions for cases where you think they'd be willing to do it but are unable to (such as a blended drink in a place without a blender), in which case it'd be most correct to say "can". You could also argue it's most polite to always say "can" so that if they say no, the implication is it's impossible, not that they are simply unwilling. These are all edge cases though, not when you're ordering straight off the menu.

The most technically correct way to say what you want to order is something along the lines of "I'd like [dish]." This still leaves open the possibility that they are unable to give you what you'd like to have, but it doesn't outright ask any silly questions. If you want to imply confidence that they are able to deliver anything on their menu, then "I'll have [dish]" is a bit more correct.

With all that said, no waiter is ever going to get confused or offended if you use "can" or "may", and some people may consider it more polite than simply saying what you want to eat.

There's an element of hypercorrection in some of this: using forms which are perceived to be "better" in some social way, even to the point of grammatical incorrectness or semantic inappropriateness. A very common example is "he gave it to John and I" which is patently wrong, but for some reason someone decided, nonsensically, that "I" is "more polite" than "me"--possibly having been corrected in childhood out of "John and me went there".

In a restaurant you can simply say "I'll have the swordfish, please"; "Can I" isn't wrong but really presumes that there may be some stumbling block, which frankly could be taken as a bit of an insult. Perhaps at a pizza place where you want one of the slices that are ready and waiting, it's a fair question. "Can I get a pepperoni slice, or are you out?"

As a guest where you are enjoying the generosity of a host, "May I" would be appropriate. I really don't think it fits the customer-server dynamic of a restaurant.

No Practical Difference

In the Pacific Northwest region of the US (to include northern Idaho, eastern and western Washington, and central Oregon), it is commonly acceptable to order food in a variety of ways, many of which are offered in other answers to this question. Here are three as direct examples:

1. "I'll have the #1..." (expressing the will or desire to have the #1)
2. "Can I get the #1 with a cola and...?" (asking after possibility and/or permission)
3. "Hmmm...get...me...uhhhhhhnumber~..." (slowly, indecisively, ordering/demanding)

MW and Ox-D each define the words to make them synonymous in this context. In all three examples above, the objective of the communication is understood by each party through context alone. Every person gets it because of where the conversation is taking place. Mechanically speaking, there is no significant reason to use either "can" or "may" as a paying customer.

Take that away and what is left is personal, cultural preference. In the US, cultural preferences generally include being treated with respect, which further includes numerous, nuanced ways of communicating. In general, it is impolite to raise your voice far louder than is required to be heard while displaying a fierce or grim countenance. It is more polite to assume a posture of submissiveness, which implies that your request is for voluntary action. Etiquette/politeness is all about submission and is egoic in nature. This makes it subjective.

If you've lived in a gratuity-centric culture and experienced a variety of food services, you are sure to have noticed that employees who earn tips put up with impolite behavior more often than those who do not. In fact, people will tolerate otherwise intolerable behavior for as long as they believe it is worth it to do so.

In summary, the tone and timing of your delivery are far more important than which word you choose.

"May" can imply that the answer can also be a "no", e. g. because of being forbidden (e. g. alcohol on an airplane, using the toilet during an exam, or something similar). So technically it leaves the answerer the option to decline the request on grounds of permissions. Of course, often this is just used to be polite without any expectation of any declining the request.

"Can" implies that one has the assumption that the asked thing might just not be possible (e. g. tap water just might not be served in this restaurant, or burgers might be out because it's too late in the evening). But again, this often is just used to be polite and to give the answerer an option of declining the request (while expecting that this won't happen).

Giving such an option of saying "no" is typically just used to enable the other person to be polite in turn just by complying to the request. Technically I could go and ask for something with the words "Please bring me the burger" or even just something like "I'll take the burger". This leaves less room for being polite for the waiter because there is no politeness in just doing as one was told to. It rather feels obedient or even servile.

But of course in many cases conventions exist that this is okay for both sides as it practical by reducing the amount of verbal exchange.

As the question is whether to use "may" or "can", I would say that using either implies that you expect that the request might be denied. Which is more appropriate depends mostly on what you expect why your request might be denied. For "may" it's more because of rules, for "can" it's more because of technical possibilities. Since you put the interpreting of the rules into the hands of the answerer, you pass more power, so "may" is more polite.

when asking to take my order

It depends on what you mean by "asking to take my order". If you want to actually order something, technically neither is correct.

"Can I... ?" / "May I... ?" / "Could I... ?"

These are questions. They would be technically appropriate if you intended to inquire about the restaurant's capacity to fulfill an order. (or your own capacity to place an order, depending on how you interpret the entire process) That's probably not your intent so you might then think of something like...

"I would like..." / "I want..." / "I will have..."

These are statements. They would be technically appropriate if you wished to communicate your desire to have an order fulfilled or a prophetic prediction. Be careful though, that might still not be what you actually mean since...

"I'd really like a hamburger, but I can't afford one."

... is also a statement which communicates a desire for an item but is clearly not a request to place an order for that item.

You're less likely to use something like...

"Please get me..." / "Place an order for..." / "Prepare a..."

These are imperatives. These are technically appropriate for orders and have no room for misinterpretation as far as I can tell but are also generally considered to be somewhat rude in this social situation, (at least, everywhere I've personally seen and in the media) probably because of how forceful they are.

In summary, you can be technically correct and kinda rude or technically incorrect and polite.

If we're going to be pedantic and insist on people using the precise linguistic form that exactly matches the speech act they are performing, then the imperative mood is appropriate:

Give me a hamburger.

This, of course, is generally considered quite rude. Slightly less rude and slightly more oblique is to inform another person of your wants:

I want a hamburger.

The bluntness of this can be reduced with

I would like a hamburger.

All other forms are simply circumlocutions, forms that communicate what you want by saying something other than what you want, but are, by social convention, understood to mean what it would be rude to say directly. Neither "can" nor "may" are, from a nitpicky point of view, correct. "Can I have a hamburger?" implies that there is some doubt regarding your ability to have a hamburger. "May I have a hamburger?" implies that you are planning on taking a hamburger, (rather than asking that someone bring one to you) and are inquiring as to whether you have permission to do so. Both use the interrogative mood for a speech act that is imperative in nature.

In the scenario of placing a fast food order, the typical employee will not be paying any attention to the nuances of the words you use. Instead, they will be listening for which item you are ordering. In that context, it makes no difference whether you use "Can I" or "May I", since the employee has already ignored all details except for "the customer has ordered item X".

At least, this is true to the degree your order is presented in a manner and with words the employee is accustomed to hear. Anything out of the ordinary might cause the employee to listen to you more closely.

(Even then, the employee might not be paying attention to the actual meaning of your words! As others have pointed out, if you use "May I" meekly, or in a region where that phrasing is uncommon, the employee is merely likely to think "the fakely polite customer has ordered item X".)

For that reason, personally, I deliberately adapt my English usage to what I observe to be common in certain settings. It is more important to me to place my fast food order smoothly than to attempt to communicate a nuance that will be lost on the recipient. In other settings, I may use more deliberate English, because nuance actually is important to me!

In the fast food setting (Pacific Northwest), I observe "Can I" and "May I" to be equivalent, so long as either is delivered casually and/or with confidence.

As for the debate, I agree that there is a difference in meaning. However, since that meaning would almost certainly be lost on the fast food employee, it would be pointless to make the distinction in actual practice. If you really needed to ask whether an item might be available, it would be better to ask directly. The employee would never assume "Can I" was intended to have the nuance you suggest.

This one ought to be recorded in the annals of overthinking.

"I'll have the burger."
You can add a 'please' or 'thank you' [implying 'thanks in advance'] which will make the exchange more polite.

Done.

May I/can I/could I/would I/should I... are all just politeness placeholders to make everybody feel more comfortable.
Use them or don't. It adds little to the overall experience.

The waiter is there to take your order. He is not there to debate semantics or to criticise your word choice. He probably wouldn't even notice.

Your need is to be polite & not demanding, whilst indicating your exact requirement.
"I'll have..." is precise & to the point. It is neither impolite nor demanding.
His need is to recognise what you want & bring it. Assuming no kitchen shortage, that's precisely what he'll do. That's his job. Assume he's good at it.
Also assume he doesn't really care exactly how you asked, so long as you didn't say "Give me a burger in 5 minutes, or else"

The usage of "Can I..." vs. "May I..." (or "Could I...") is simply an argument of capability vs. permission. In practice they are often used interchangeably and the waiter knows what you mean, so it doesn't really matter.

For the grammatical elite, you should use "Can you..." because...

1. You are not capable of getting the food yourself.
2. You are not asking for permission.
3. You are asking if they are capable of getting the food.

"Can I..." vs "May I..." Example:

Student: "Can I go to the restroom?"

Teacher: "I don't know, can you?"

In this case the student may have some reason to need assistance in going to the restroom. This is not clear if they are capable or asking permission.

Student: "May I go to the restroom?"

Teacher: "Yes, you are excused to go to the restroom."

In this case it is clear that the student wants to leave the classroom to go to the restroom.

Can and may cannot be used interchangeably. May is used for permission. Can is used for ability. Here are some examples:

May you give me the menu?

Technically the waitress can give the menu — it's an ability. If you were to use can, it would have a completely different meaning. It would have been questioning whether the waitress has that ability.

May I use the restroom?

Here you are likely asking for permission — not whether you have the ability. If you were to use can instead of may, it would have a completely different meaning. The sentence would have been about whether you have the ability to use the restroom — that's likely a given.

When speaking in first person you should never say “can I …”. Can should only be a statement in first person. For example, I can play the piano. For a student to say: “May I go to the bathroom?” Is correct. In summary, may I is asking permission sometimes, but at a restaurant saying “may I have a cheeseburger, fries and a coke shows very polite manners and it is technically the correct usage of English grammar.