How can I test my house water supply for hydrocarbon presence?

This question might be better suited for Chemistry or another place, and I'm not certain Earth Science is the place for it.

Without being an expert on groundwater, water supplies, or hydrocarbons I would say this.

First of all, "deteriorating water pipes" should not lead to any smell. Certain trace metals may be leached from metal (mostly copper) tubing, but you will not smell them. Plastic tubing (mostly PVC) likewise does not cause any smell.

What you are smelling could either come from the water supply itself, and then it's a municipal problem. Hydrocarbon contamination of wells or aquifers maybe? Have you raised your concerns with your local water supplier? Are you the only one who notices this, or do you neighbours witness this as well?

Testing it yourself would be hard. If the concentrations are high enough, you could ignite them. A safer way would be to pour the water to a bottle. Do you see any oily film on top? Does this change when you refrigerate the water? When you freeze the water?

The fact that you smell it, means it is already volatilising out of the water when you open the tap, so whatever sample you take of the water will underestimate the hydrocarbon contents.

You should also consider the fact that it's not coming from the water. Possible that it's coming from the outside, and it just happens to occur (or you notice it) when you open the tap.

While the best way to test for hydrocarbons in drinking water is to get a specialized test kit like the one you posted in your edit or to send a water sample to a lab, there might be a cheaper DIY way to do it. I haven't tested it but I think it is worth a try:

Detecting hydrocarbons is exactly what is typically done when detecting leaks due to broken head gaskets in combustion engines. A broken seal in the combustion chamber of the engine can push some vaporized gasoline into the coolant. Therefore, the standard way to find such leaks is to test for hydrocarbons in the fumes that come out of the coolant when opening the radiator cap. To do so they sell very cheap tests that allow you to suck up those fumes and get them to mix with a chemical reactant which changes color if there are hydrocarbons present.

Here is an example of the test:

And in this video, you can see how it works.

If you Google "head gasket leak test", "combustion leak kit", or "block test" you will find tons of options, and you will very likely also find it also in local shops.

However, the concentrations in your tap water would be much lower than in the design application of these tests. Therefore, the detection that takes minutes in a car might take hours in your tap water. I envision that you may be able to leave the tap open with one of these tests continuously sucking the fumes coming out of the water (if any). To do so, you can put a tube from your vacuum cleaner to one end of the test device (adjusting the size and leakage of the tube so it provide a reasonable amount of suction), so it will get the fumes to bubble through the liquid as long as you want.

This is a very rough hand-drown illustration of the setup I'm describing:

If you get it to change color, then you will then need to figure out the concentration. That would require some additional tests, perhaps doing the same with water with a known amount of hydrocarbons added. Alternatively, you can use that evidence to decide whether to go forward with a professional testing or not.

Chances are you do not have presences of fuel in your water. This sounds like you could have a few different things happening.

First off, iron reducing bacteria can lead to significant corrosion. These are bacteria that feed on iron both in the water and also in plumbing systems. They also can put off a smell.

The second one is manganese. If you notice this smell after rain, etc. then this is a possibility. Manganese can make the water blackish and smell like fuel oil. While mostly harmless, it can cause issues with staining, etc.

If you have a private well and live in a hard water area, both of these could be occurring simultaneously. Correlation is not always causation. What the smell is vs. what is causing the corrosion could be separate issues.

Lastly, if the pH is off in the water, too acidic or too base then this can also eat away at your pipes. Typically though, many water quality smell and plumbing issues come from bacterial issues. Check this first and then move on if negative.