I heartily second what Allison has already said.
Also, consider thinking smaller about sponsorship. Don't start out thinking Ariat or Purina, think local. Ask your local feed and grain dealer, tack shop, hay farmer, riding school, etc. if they'd be willing to pay entry fees or something similiar in return for an endorsement. Ask a local purveyor if they'd like to donate a custom saddle pad, monogramed choker, silver belt buckle, etc. for you to wear in the ring.
Finally, my favorite creative sponsorship arrangements were the one proposed by my boarders and clients - they paid my way into a bunch of clinics with big names. Their return was free auditing of the clinic and then a post clinic lesson or seminar with me. The local Pony Club also sponsored me in several clinics and competitions, in return, I had to teach a Pony Club clinic or seminar. Great arrangement that paid multiple dividends.
Ultimately though, I go back to the point that Allison already made - concentrate on building your skills and resume, and the rest will follow.