I know that UA does not allow this - the dog fee would count as a baggage or cargo fee (depending on how you ship them), and UA only accepts actual money for that. I don't know of any U.S.-based airline that allows mileage payments for such fees.

However, given that this is a charitable act, your friend(s) might consider calling up some airlines (their Customer Relations or WHQ, not regular Reservations, of course) and asking about a special exception or a "charitable donation" of services.