If you have ever participated in a youth sports organization of any kind, you know there are expenses involved in running a successful program. This it true for Horseshoes too, however the costs are not nearly as a high as more common sports such as football, baseball, soccer, etc. The donations received by sponsors can be used to purchase supplies for your league (water balloons, bocce balls, corn hole, Horseshoes, stakes, PVC piping, shovels, brooms, shirts, treats, prizes, awards, food for BBQ, etc.). Fortunately, most community business are very supportive of youth organizations and will sponsor without too much asking. Persistence is key, however, once you contact a potential sponsor, keep in regular communication with them until they commit to, or decline, a sponsorship. If they indicate they will get back with you, ask for a timeline and a business card. When you are turned down (some will), don’t sweat it, just thank the business owner and let them know you’ll be back next year. Persistence is Key.
The amount of funding you should pursue depends on your needs and goals, which will certainly vary by league. A general target might be to raise $100 per participant. Depending on your costs, this amount may allow your league to purchase horseshoe for the youth. Horseshoe manufacturers may offer discounts for bulk purchases, contact them to find out.
Most sponsors are interested in receiving some kind of 
recognition, so be sure your sponsor packages offer this. Business names on the youth shirts, a banner at the courts, and a thank you plaque they can display are traditionally well-liked. Take care of the sponsors who are taking care of your youth.
Because most Americans are familiar with horseshoes, you might hold a doubles tournament for Sponsors and award them prizes and recognition too. This is popular where it has been tried and produces long-term sponsors who come back year after year.
Sponsorship request letters and techniques are abundant online, with a quick search you will find many. This one has been shown effective with youth horseshoe leagues. The amounts you request can vary. You might be surprised at how willing your local business are to donate large amounts of funding.