Here's the information on the pediatrician my friends used in Hilton Head:
Peter J. Salerno, M.D.
Salerno Pediatrics
35 Bill Fries Drive Building I
Hilton Head Island, SC 29926
843-342-7337
As far as insurance goes, it will probably depend on what you have. I know BCBS is pretty easy to use in other states with a bit of research first (BTDT). Some doctors you may have to pay up front then get re-imbursed from your insurance company directly, according to your plan's limitations and deductibles. BTDT as well.
The safest thing to do is take them to the doctor before you leave if you are concerned that they have a lingering illness. Try for 48-72 hours ahead of time, so if they need antibiotics for an ear infection or whatever, it has time to work. You could also touch bases with your doctor this way, so if they get worse, you can call the office, tell them what's going on, and just have a prescription called in for you where you are (BTDT too!!!!). Sometimes mother's intuition makes the best diagnosis though....
When are you going? We'll be in HH May 26th, spending the night on our way to WDW. Can't wait!!!!