I was a lot like you! I didn't start really riding until I was 18. I got my start volunteering at a therapeutic riding center, then got a job at an equestrian kids camp, now I'm 24 and own a thriving riding academy. I'm booked to the max with 17 students, 5 horses in training, 7 people on my waiting list and 2 more asked to be added to my waiting list today. I'm hiring someone to help whittle down the waiting list. I never had the opportunity to do FFA or 4H or anything either. I must attribute my success to God, in every way. He's the reason it's all working.
Like others have said, network. Start making friends with people at the barn. A lot of horses go under-exercised, you might land an exercising job. Treat every person you meet like you want to be treated. Consider moving out of the area to find work. Be a camp counselor at an equine camp. You'll get there if you want to!