Mr. Prole said that his country is on the list, which means that they should offer 1949 IDPs. (It would have been helpful to name the country unless you are particularly embarrassed.)

License holders from some countries (one being Germany) that did not sign or do not offer 1949 IDPs are able to drive without an IDP but need to get a translation from JAF. This is convenient for people who move here, but a bit of a hassle for a tourist.

In my experience, the rental car companies do not even ask for the real license but only scrutinize the IDP and passport carefully.

Actually, that list from the police is misleading because some European countries that signed 1949 do not issue such IDPs. (Presumably F and B). So those countries also allow driving with a translation like Germany and Switzerland, which did not sign the 1949 convention. It says that you can get one from the consulate, or you can get it from JAF when you arrive. Check info from Toyota:

https://www.toyotarentacar.net/english/guide/license/

p.s. Car rentals in Okinawa appear to be different -- not the usual corporate outfits that you see on the mainland but mom and pop businesses.