[schroth-sensei]

Like Josh has stated, I wasn’t able to find a rental for a shower chair in Japan (hygiene issues essentially require them to sell them new, which is probably for the best when I thought about it). Also, purchasing one locally in Japan can be very expensive or problematic (many places may not have local inventory available for immediate purchase, limited selection, or like Japan’s amazon website may have sellers from other countries who don’t immediately ship the product when you request and need it).
Probably the least expensive to fit basic needs may be a collapsible camping/outdoor chair, some are quite portable. If such a thing will work for you and your needs, it may be possible to get one small enough to fit into smaller shower areas you often see in Japanese bathrooms.
If you have found a hotel/rental with a bigger shower space, then it is possible to buy and bring a bigger collapsible shower chair, my suggestion would be something like the one seen here:
https://shower-buddy.com/mobility-products/eco-traveller-sb7e/
I am not paid or affiliated with them and nor have I personally used this chair, but I do use a shower chair from this company that is very much like this (just not collapsible), it is sturdy and works well for my needs. The one I linked above is fairly inexpensive (last I checked) and they offer optional accessories including a travel bag for it (Note: If a US flight, I cannot say for certain that it will fit or have to be checked-in, but this should qualify as a free carry-on because it is your medical equipment). I have seen some other similar products elsewhere but they can get expensive, nevertheless I suggest doing some research before making any decision.

I hope that help,
-Justin