Even if they own it, you can contact them and ask if they mind you selling a modified version of it.
I'll bring it up then and see what they have to say. Because I'm planning on redoing a lot of the code and adding some features, I was planning on giving them the new improved version for free. They win either way, they just had to pay more up front.
Rather, a better strategy would be to talk to a legal professional who understands the law in this area. Because the contract did not spell out who will "own" the code, he (the author) must gain an understanding of the legal perspective in these situations before he squanders his opportunity to reuse the code.
Even if the conversation with the client resulted in their agreement that the bherms can do whatever he likes with the code, that doesn't clear up the legal issue. What happens a few years from now when this charity is under different leadership and they notice the similarity between their management system and the one that sells for $30/month from whatever business bherms starts?
Talk to someone who knows these answers before you proceed. Nothing is worse than putting years of effort in to something, only to have it dismantled by something from your past that you could have taken care of easily.