My 10 year old daughter takes care of friends' and neighbors' pets when they go away. Of course, I have to drive her back and forth a couple of times a day, but she does all the work to earn the money and it's mainly during the summer. She also saves birthday and Christmas money she receives from relatives.
Babysitting and stable work are great ideas, too, so it is feasible for you to earn enough money, say, over the course of a year to buy a horse, but, as others have suggested, you have to look at the bigger picture. How would you manage board, vet and farrier bills?
My girls were born into horses so it breaks my heart to see young girls who have been bitten by the horse bug, but whose parents are not as enthusiastic. I hope you're able to work horses into your life somehow.
(Side note: I was driving the truck in the hay fields at age 12, too.