I think this is a terrific idea, and I've done similar things sub rosa. But at the risk of being a big skunk at the picnic, I'd say this needs to be examined with reference to individual FF program T&Cs regarding gifting or transferring of miles.
Although it's with entirely different (and laudable) motives, this sounds a bit like a mile brokerage model, depending on how you handle the actual acquisition of the tickets. AFAIK, most (some? a few?) airlines won't let you donate miles except to their own designated recipient organizations. Transferring the custody (and not ownership - the airlines always own the miles) to a third party who then uses them to purchase tickets might be in violation of the T&Cs.
Now you could be a matchmaker who contacts the donors who've made pledges and have them acquire the tickets for the recipients, but that would eliminate the anonymity of the donors, and the donors would have to pay the fees and taxes themselves. You probably wouldn't be able to reimburse the members for this, and neither the value of the miles nor the taxes/fees would be tax deductible for the donor because it wasn't a 501c3 recipient. (The miles aren't deductible anyway since the T&Cs all say they have no cash value.)
Like I say, not trying to be a poop, but I'd check some of these things out, maybe with an anonymous phone call to a major FF program to see how they handle these kinds of requests. (You wouldn't have to mention refugees, you could ask about donating miles for families of sick kids, for example.)