AFAIK, you can propose yourself. I don't know of any self nominated MVP, but, basically, I see no difference if it's done by you or anyone else.

My advice: don't do it for the benefits, cause you may be disappointed. I mean: even if you get the MVP, if you do it for its advantages only, you may end up thinking that the time spent was not worth it. So, go for it only if you believe you can do the same thing as if no MVP award were involved. Or, better said: Will you still be happy if you don't get it? If yes, go for it! :) Don't think about the MVP benefits as about a pay for your work! :)

Vlad