Call around to other barns to see if they will trade work for a lesson. You can always politely ask your neighbors, but be aware that they might not want you riding their horses for various reasons. Maybe their horses are too much to handle for a beginner. Are there any therapeutic riding centers or horse rescues around you? You might not be able to ride there either for a while, but at the very least, you might get to help groom the calmer horses and work your way up.
I'm not sure if writing a letter is that helpful in today's society. If I got one, it would just get put to the side and forgotten about (this might vary depending on what country you are in). If you can find a phone number, I would call up the barn and talk to someone. Or for your neighbors, see if you can catch them while they're out riding or stop by some time for a chat.