Where is the closest grocery store?

Brunswick

Best to get groceries before you arrive. But if not there is a Winn Dixie if you leave the island, go across the bridge and its not to far out on the left.

There's a little commissary store on Jekyll where you can buy food but if you're looking for a bigger grocery store that offers more variety you'll have to drive into Brunswick. I'm estimating about 15 minutes to get to the bigger grocery store.