This article is not exactly meant for your situation, but includes the unconventional tip of using an American Airlines Citi card to buy absolutely everything to earn a “free” flight.
This guy highlights the best promotions at the moment.
Both of these sites target an audience of frequent travelers, but with some adaptation, you should be able to get a pretty cheap flight.
I would also suggest that you consider going with a church, like the others have said. I’m not religious myself, but the mission trips my friends have been on have not been particularly religious, and they got to go for about ½ the price of comparable programs. Is your preference for a secular opportunity worth a couple hundred or even a thousand bucks? It also takes care of having to find a volunteer opportunity yourself.
Also, depending on your reasons for wanting to volunteer, you may want to consider volunteering locally. If you just want to do some good or add to your resume, then volunteering locally is at least just as good, if not better, precisely because it is so expensive to volunteer internationally. Honestly, you may be better off simply donating the money would spend directly to a charity. However, if gaining some international experience is important to you, of course, it might be well worth the expense.