We have participated in a number of mostly self-funded service trips under the umbrella of a nonprofit. For us, it worked out best to have the nonprofit purchase the airline tickets (no real discount that I'm aware of), using money that we donated to the nonprofit and then earmarked for the trip.

Doing it this way at least gives you a tax deduction for the money you're putting into the service trip. OP's daughter, being fresh out of school, may not be in a high enough tax bracket for this strategy to make a significant difference. But for those in a higher bracket it can save 30+ %. Every penny helps.