No medical reasons, no I don't know anyone there, no I've never been there, no driver's license.
I know they can put me where they would like to, but I'd like to select three places that would be suitable on the offchance that I get a placement I request, and I'd like to be able to justify it when they ask why I selected those places.
While it isn't impossible to get around without a car (particularly if you are a strong cyclist, have lots of friends with cars, or are lucky enough to be in a large city with public transportation), I strongly advise getting your driver's license if possible. Getting an international license is a simple, $15 process and attempting to get a license in Japan after the fact with no home country license is a long and costly process. You'll be spending money and vacation days (since the DMVs are only open during weekdays and during school hours) attempting both the written and driving tests, and the easier you can make this on yourself, the better. Something like 85% of foreigners fail the driving test the first time in my prefecture, largely based on counter-intuitive rules and wildly varying instructors.
That said, even in my country town, there have been JETs who didn't need a car and got around fine. This is a decision made AFTER getting here, though. They also had to ask for rides to every monthly meeting, every demonstration class, and every BOE or school run party. It's not that bad if you're close to folks in your area, but being at the mercy of someone else isn't a feeling I think many enjoy.
I also find a car to be extremely helpful as a JET. It snows often here, I'm constantly toting teaching materials between 3 schools, and it offers a great deal of freedom. That said, the "every situation is different" JET mantra will be drummed into your head if you get in, and you may find that none of this advice applies to you.
In terms of city size, you can request away, but it's all a crapshoot. Hope you get in, though, and find a placement that works for you.