I don’t have any answers for you, but after reviewing the web page you linked, I gotta ask - are the fees they charge normal for “volunteering”? Roughly speaking, they want $1500 for the first week and $1000/week for the next, then it goes down to ~$300/week. That does not include airfare, but does include room and board.
That seems a pretty penny for the privilege of volunteering. Am I just out of touch?
Look at the book Volunteer Vacations. You might find something there. In addition, typing “Volunteer Vacations” into Amazon’s search bar on the main page yields several more books of this ilk.
I’m not sure, but I think for many programs paying around this much is a standard. They provide room and board, medical insurance, clean water, and so forth. As well the money goes towards helping their programs survive. A friend of mine is considering a volunteer vacation as well and he told me lots of programs charge this much.
A good friend of mine is a teacher who is participating the the Fulbright Teacher Exchange. It’s not for the summer, but for the school year. He will get his salary plus living expenses in his host country. You might also do a search for programs sponsored by the State Department’s Bureau of Economic and Cultural Affairs (BECA). BECA funds a variety of exchange programs. In the former Soviet Union, you could look at something like ACTR, although their programs might all require Russian language skills, they work with a lot of partner organizations who place volunteers, they might be able to direct you a good program.
In evaluating programs, look at who funds them. Some programs are sponsored by churches and might have a religious element that you may, or may not be comfortable with. Other programs might be poorly supported and provide subpar logistical support.