Sounds like poor planning on the part of the birthday child's parents. It's not a charity, so what is it? It doesn't accept donations, so exactly how much investigation did the parent do?
Is the party this weekend? I'd either send my child with a card and maybe $10 inside, and leave it to the family to donate that cash or not. You could include a note saying there's no way to donate. I'm assuming you have already considered sending a check and that you're not insisting on an on-line donation only.
My big concern though, is that this is somehow not a legit organization and therefore you wouldn't want to support it, or it's a municipal agency funded somehow by public dollars which is not used to getting (and perhaps not able to accept/process) donations. Without more answers, it's hard to guess how to advise you.
If the party is not this weekend, you can wait for the organization to contact you, or you can contact the family and ask for more directions on how to contribute. But if they tell you, then you kind of have to do it, and I think it's important to find out what the group does before you part with your money.