Hi Kevin!
Welcome to the Disney Parks Mom Panel!
While the Walt Disney World property maps make it appear the distance between Disney's Caribbean Beach Resort and
ESPN Wide World of Sports is small, it is actually about three miles away and crosses a major highway (Osceola Parkway). However, since you are staying at a host resort, Disney Bus Transportation is available. The buses run one hour prior to the sports complex opening until 11:00 p.m. or when the complex closes, whichever is later. These buses typically run on the hour and 1/2 hour.
If you feel this schedule will not fit your needs there are a couple of other options. As you suggested, you could rent a car. Since you are a Disney resort guest parking is free at all Disney theme parks as well as ESPN Wide World of Sports. You would also hail a cab. Most cab rides within Disney property cost less than $20 including tip.
I hope this helps. If you have any more questions please come back and see me.
Good luck at your competition. I will send loads of pixie dust your way to make your gymnast fly!
Sue N., IN