Oh no - that isn't the thing to do.
There's a problem with writing to a company; they can't pretend to know nothing about you.
Specifically, they are *going* to say no. There's no way they'd say anything else; even if it's non-profit, they can't let people use their property, otherwise *everyone* would start doing it, and point at *your* site for justification.
And more to the point; once they've said "no" they're pretty much obliged to shut you down. An awful lot of fansites are illegal in some way; fanfic, for a start, breaks copyright all by itself. So long as Square isn't specifically told about them they can always pretend they don't know (and you won't be surprised to know they don't go out looking for fanfic sites to sue) - but if your site is drawn to their attention, then they have to defend their copyright.
