While here in Okinawa, it is often convenient to order things online from Amazon, but using the US site is not so convenient (long time to ship, high shipping fees or not shipping overseas at all). So the answer is Amazon.co.jp.
You need to make a new account since your old Amazon account will not work (you cannot just switch countries). No problem. Some things will be in Japanese, but a lot of the shopping screens and menus can be shown in English (just switch the language setting in the account to English), so it is actually quite easy and no real need for Google Chrome to translate.
For delivery options, you can have it sent to your address in Japan (sometimes you can even set the delivery date/time ahead, otherwise you may need to scan QR code on the slip for re-delivery), but it is also possible to have it sent to the nearest convenience store (such as Lawson or FamilyMart). You can not only pick up at the conbini, but you can also pay for it there. Talk about convenience.
As for payments, foreign credit cards are accepted (for everything except for Digital music). Video, regular items, books, and everything else except for Digital music can use foreign card! Otherwise, you can also utilize cash on delivery (to your house or at the convenience store). Yes, this works in Japan, I have done this before. It sort of amazes me. The drivers even carry change if you have it sent to your house or apartment.
As a student, I qualify for Student Prime membership, and it is only 1900円 per year. There is also regular Prime membership for only 3900円 per year, unlike the US which is ~10$ a month. You can access Prime video, Prime shipping, and special deals. To use Prime Music though you must have a Japanese credit card . So keep this in mind, as this is the only frustrating thing for me.
Amazon Video has many movies and TV shows in English (with Japanese subtitles). If your Japanese is good enough, you can watch the regular Japanese movies and TV as well (but then again, you probably are not reading this if your Japanese is so great). It has just about everything the US Amazon Video has, plus some. Titles in foreign languages generally only have the option to subtitle in Japanese not English, so you may miss out on somethings. But it does have all of the old season of X-Files, so that’s pretty awesome.
Oh, and I should mention… you earn points that turn directly into cash discounts when you order from Amazon.jp. Often you can get bonus points, just for signing up or certain items will have extra points attached. Really, it is a pretty great service, and when I realized how cheap the Student Prime membership was through Japan, it has definitely been worth it. I have purchased books, a camera, kimono accessories, and even liquor from Amazon.jp… often times it is cheaper and there is more variety than buying from the department stores.