I'm in your same boat as far as being a nonprofit applicant, though my goal is to transition into the for-profit sector. I actually don't think of Chicago or Penn being particularly strong in social enterprise (the umbrella under which b-schools tend to classify nonprofit management). But I could be wrong.
Harvard, Kellogg, Haas, Yale are all well-known for social enterprise. Stanford is also strong in this area, but moreso for social entrepreneurship. And I've heard that Johnson/Cornell is up-and-coming in this area.
The Net Impact website is a great resource. It has a publication that describes the social enterprise environment at every school that has a Net Impact chapter. And there's a prominent ranking of MBA programs based on strength in social enterprise called Beyond Grey Pinstripes.
From what I hear, MBA programs are pretty open to nonprofit applicants (though I have yet to learn that myself -- still awaiting decisions from the programs I applied to). I think your bigger handicap will be your age. More and more full-time MBA programs are trending younger (25-28), and they view your age in terms of how old you'll be at the time of matriculation, not at the time that your application is submitted. So you should started thinking about how to best spin that now.
Another typical challenge nonprofit applicants have to deal with is convincing adcom that they have strong quant skills. A good quant score on your GMAT is the ideal way to convince them if you haven't taken any quant-related courses or don't have quant-related responsibilities at work.
Good luck!