Hi there
I would divide up the purposes of the wheelchair permit.
There is an official permit that requires you to take your Japanese disability ID to the police department to apply (taking a few weeks). You get a special sign to put in your windshield with an erasable marker so you can write where you have gone in case the police etc want to ask you to move. But that is because the permit is for letting you park on the street in otherwise ticket-able areas (ie on a small road in front of a store).
Parking spaces at malls, convenience stores etc with a parking lot are on private property and are a requirement for the owner to have, but unenforced in terms of getting a ticket for parking there without a permit. In fact, most people who need it find it in use by a driver who doesn’t need and the government’s only recourse has been to create a poster campaign saying “please don’t park there if you don’t need it!”
So, as a visitor without an “official” permit, avoid parking on the street, or other no-parking areas etc. But, there should be no issue parking in a parking lot’s accessible parking space.
I would bring your permit (or maybe a color copy), or some sort of wheelchair sign, to put in your window to let others know you are parking there with a reason.