I think it would be a very bad idea getting an untrained youngster for your first horse. Especially since you'd have to wait about a year or so to ride her anyway.
Honestly, I had the very same problem when I was 13.
What did I do? I went to the stable I took lessons at, started volunteering, and eventually got a job. (Yes, at 13!) I didn't make a lot, but I made something, and it was better then nothing.
SOOO many benefits came with working at a stable.
- cheap/free board
- can ride any horse
- can use all equipment
- spend more time with your horse
- get to go to auctions
- got to see MANY natural horsemanship clinics
- got lessons by many trainers
- met many wonderful horse people (vets, farriers, boarders, etc)
the list is endless. Try to work at that stable you take lessons at.
Maybe sometime they'll take you to an auction, or when you do have enough money to afford a horse they can take you to see it. Which is a big help because they can help you decide on weather or not the horse is right for you. Horses are going for $25 dollars at auctions right now. Just save up, have a little patience, and eventually you'll find the horse that's right for you. Don't feel like you have to rush.