Hi there! I detect here in Japan, but Okinawa is pretty far from Yokosuka so even though I do speak enough Japanese to ask for permissions I can't really help you out in person. Got a possible suggestion, though.
If you happen to be US military (not many westerners know of Yokosuka unless they're military) then maybe your base has a translation service? I'm not actually military myself, but I listen to AFN radio here sometimes and I've heard such a thing mentioned.
If you have access such a service, it might be helpful to write out a polite, generic request ("Hi, I'm a hobbyist looking for X type of things with my metal detector. With your permission, I'd like to detect on your land. I promise to be respectful of your property and not leave any holes or trash. May I detect on your land?" or something to that effect) and then have it translated into a document that you can copy and show to a landowner whose property you'd like to detect on.
That way you only have to learn a few phrases (sumimasen-- pardon me, onegaishitai koto ga arimasu-- I would like to ask you something, etc) rather than attempt to converse about your hobby in a language you don't know.
I would offer to translate a letter like that for you, but my Japanese is pretty coarse and if you're making an unusual request like this (since detecting is not a common hobby in Japan it is unusual) you'll want to do it as politely as possible.
As for whether you will be given permission...i have no idea. I just work beaches, which are more or less public domain. Detecting on land in Okinawa is a bit risky. Developed land has to be scoured for UXOs before it can be built upon, so there's little of historical or monetary value to be found there (aside from recent stuff), and undeveloped land...well...they called it the Typhoon of Steel for a reason. There's all kinds of gnarly stuff out there. I geocache out in the jungles and have found some scary stuff-- no detector needed.
But yeah, if you're very polite and have either a written request or a local friend to help you ask, you may well be able to get permission to do some interesting hunts on the mainland.
