What the posters say is all correct.
1. Look up what the commercial practices laws are in your state. Some provide for treble damages, costs, and fees. Usually you have to send a written demand to your targeted company before you can make that claim.
2. Make the demand in writing, certified mail. Find where the LLC is registered and who their agent is to receive "service of process." The corporations secretary for your state should maintain a website with that information. Send the demand letter there. In the letter, explain who you are, the race you won, the amount you are entitled to, and what will happen if payment is not made within 30 days.
3. If you do not receive any response, then serve them with the summons and complaint. It will be sent to the same address in Step 2. And send the copy of that to the local papers/publications in the area, explaining that you got stiffed. No company wants a news story coming up in Google about fraudulent practices........
I have heard of larger races doing that to runners before. Sometimes it's because the directors don't have their act together. But regardless, no excuse if they've received your communications and haven't responded.