I think writing a letter or just calling and leaving a message is a great idea!
Just introduce yourself to your neighbors, tell them where you live, and that you are interested in learning how to ride. Let them know that you are on a low budget, so you can't afford lessons like the other stable offered, although you did try to see if you could work off lesson expenses there. Let them know that you would be willing to work in exchange for riding their horses, and getting instruction from them. You'd have to come up with some sort of work/riding schedule.
Just know that horse chores aren't easy, and they do require an hour or two each morning and night (depending upon the number of horses they have). You'd have to plan accordingly, and be willing to help do the "crap" work (as I call it).
Sounds like a good idea. Just be honest, and I'm sure you'll do fine!