Well to get big sponsors you have to be a well kown rider or "up and coming star" is what they call them. I got my sponsor from asking around. But to start, ask local businesses, make up a paper which professionaly states that you would like them to sponsor you for so and so show. Don't ask for an amount take what they give you. And when you get sponsors a nice thing to do is make a sign with their logos and hang it by your show stall. Always said a thank you note and a photo of you and your horse with the sign. This makes them look at you as courtious and they are willing to sponsor you again. Its tuff to find a sponsor that will keep sponsoring you. It took me a while to find a solid sponsor. Until I did I went off of many local businesses all donating different amounts of money.
I hope this helps and good luck! I am sure you will find a good sponsor!