Best way to buy Japanese yen for travel?

Have you tried calling a Forex broker and asking them if you can take delivery on currency? Their spreads are likely to be much lower than banks/ATMs.

Unless you need extremely large sums of money, I suggest you use an ATM or look for a credit card that has no foreign transaction fees (rare).

AFAIK, it's not possible for a retail buyer to purchase currency at the current exchange rate quoted online. You are always going to be paying some spread above that, and the ATM gets you the closest.

You could also try to use a bank that has branches in your country and Japan (like HSBC) and do your banking there. Then you likely wouldn't have to pay as much in fees (and possibly could draw on your account in Japan).

You don't. When you get to Japan, use your ATM card to withdraw local currency. My bank (ETrade) doesn't charge me int'l fees.

When I went on vacation to London a few years ago, I looked around at banks with ATM deals with UK banks. I found that B of A had a deal with a UK bank that you could use their ATMs to take out money from your US account for practically no fees. So the week or so before I left, I opened an account at B of A, put a bunch of money in it, and used the B of A debit card during my trip as much as possible.

Check whether you're being charged a "Cash advance" fee with your withdrawals, because it's being withdrawn from your credit card account.

If that's happening to you, then having a positive balance on your credit card account will dramatically reduce the fees. Quoting from my answer to a similar question on Travel Stack Exchange:

It turns out that even though "Cash advance fee - ATM" has "ATM" in it, it doesn't mean that it's being charged by the ATM you're withdrawing from. It's still being charged by the bank of your home country.

And depending on your bank, that fee can be minimized by having a positive balance in your credit card account. This isn't just for cards specially marketed at globehoppers and globeshoppers (mentioned in an answer to a similar question), but even for ordinary credit cards:

Help minimise and avoid fees

An administrative charge of 2% of the value of the transaction will apply to each cash advance made on your card account, where your account has a negative (debit) balance after the transaction has been posted to it. A minimum charge of $2.50 and a maximum charge of $150 will apply in these circumstances. Where your account has a positive (credit) balance after the transaction has been posted to it, a charge of $2.50 will apply to the transaction. Any such charge will appear on your credit card statement directly below the relevant cash advance.

A $2.50 charge if your account is positive, versus $20 if the account is negative? That's a bit of a difference!

I already commented the best existing answers, however let me note a couple of other things. Some of my friends in the past have wanted to do one of the following:

1. Bring cash - This is not necessary in general, though I would bring some cash in case your ATM card doesn't work. (Strangely a lot of cards from Korea, particularly Hana bank do NOT work in Japan). If your ATM card has Cirrus or Plus logo (Most US issued cards do), it should work at JP Bank and Citibank ATMs, and probably most 7-11 ATMs. Bring only $100 - $200 USD worth of cash to tide you over until you find an ATM. (For a small amount like that, you can convert cash at the airport since despite their high margins, your loss will be minimal). The point is - don't bring wads of cash to convert, or convert wads of cash before coming unless you have access to some magical business that will convert your cash with no profit. The ATM is you best bet for converting cash in most cases. Bear in mind that if you convert too much cash before hand and have to convert it back, you will pay conversion fees twice!
2. Use Traveler's Checks - This is not only not necessary - it's inconvenient in many cases. Most Japanese banks will just look confused if you present them with Traveler's Checks, and although JP bank will convert them, it requires finding a branch that is equipped to deals with them, going there in person, filling out paperwork, and waiting. Other than that, the main places equipped to deal with them are the larger fancier more expensive hotel chains. If you are staying at one of those, you probably have everything figured out already anyway. If you aren't staying at one of those, they won't want to exchange your traveler's checks. At the end of the day, traveler's checks may buy you a limited amount of safety over carrying a huge amount of cash, but again if you can use your ATM card, that solves the problem in a more convenient way. If you have a Japanese bank account, you can use a service like Skrill to transfer money with relatively low fees. If you have an international credit card brand (Visa, Mastercard, American Express, or JCB of course), you can almost certainly use it in Japan so long as international purchases are enabled.

Personally, my solution for this is to use a Transferwise borderless account and debit card (available since 2018). Although not free, rather than adding a load/spread on top of the mid-market rate, they quote the live mid-market rate and then add an explicit fee, which is usually much smaller than the banks. For example, if you were to add 1000 AUD to a Transferwise account and then convert them to JPY, you would pay a fee of 10.32 AUD and get 75,247 JPY at an exchange rate of 76.0319 JPY/AUD, which is the current live mid-market rate. You can then use the debit card associated with your account to take out cash (without extra fees within relatively small limits) or pay within in Japan (without extra fees).

I have personally used such a card for USD, GBP, EUR, SEK, NOK, CHF, and RUB, and it works well for me. You might be cheaper off only with a bank card that truly charges no foreign exchange fee and does not add a load/spread on the mid-market rate, but those may be hard to find (see Is there such a thing as a credit card with no (currency exchange) fees?).

An option for you to look at for potentially free (or much cheaper) ways to buy yens is Revolut, that recently launched in Australia.

You can withdraw money at an ATM in local currency using your account for free or at much lower fees than the ones you quoted depending on the account type you opt for. You can also pay using your card and be charged just the intrabank rate of exchange without fees at all. Given Japan is primarily a cash-centric society, it would make sense to have some yens in hand, but overall you should still be able to use your card in most transactions and benefit of the no-fee exchange rates.

That and Transferwise that @gerrit mentioned are most likely the most cost-effective solutions for what you want.