Thanks to one reader, I can report that you can get a gluten-free burger on Amelia Island.
The Falcon's Nest, a bar and restaurant at the Omni Amelia Island Plantation Resort, does offer a hamburger with a gluten-free bun, according to its menu. It doesn't say if the french fries are gluten free.
That's a long way to go, for me anyway, to get a burger. But you may want to check it out.
You can find more information and a link to the menu here: