We're planning to launch our app and want to give it away for free to non-profits so that:
Does anyone have any experience with attracting non-profits to your product? Since we won't be charging them, I would guess many would take us up on the offer.
How should I go about getting started with contacting non-profits?
If you're giving your app away to non-profits for free for giving back to the world, that's great! If you're doing it to drive sales, that might be a stretch.
I think sometimes the idea of free seems easier than selling it but in reality it is sometimes difficult. People don't necessarily value free like they do something they pay for. And although it is free, you still have a "sales" job to do with the non-profits to show them the app makes sense and is worth their time.
I recommend you have a story or some logic as to why you're giving it out for free and ask for something in return of equal value (i.e. customer testimonial, a video of a happy user, a beta tester, etc.).
Quid pro quo -- this for that. You give them the app and you get something in return (that is documented and part of the transaction for them).
Chris makes a good summary, but, reading:
"How should I go about getting started with contacting non-profits?"
Suggests that you haven't spoken to the non-profits, in which case, what makes you think they will want your product?
If you know why they would want your product, you can determine who (department & manager) wants it and then make direct contact to the appropriate people.