One great thing about Disney World restaurants is that being gluten free is almost never a problem- most of them have plenty of gluten-free options (the only exception is some of the World Showcase restaurants at Epcot). My son is gluten free & he loves eating at the Disney World restaurants. If you google "gluten" along with the name of a WDW restaurant you're interested in, you'll find reviews of Disney restaurants by a gluten-free reviewer. The reviews can be very helpful.
One thing that is a bit of a splurge but that can be great on arrival night is the Hoop Dee Doo Revue dinner show at Fort Wilderness. To get there (since you have a car), you'll drive to the front of the Fort Wilderness resort, then park and take an internal Fort bus to the Settlement area where the dinner show is at Pioneer Hall. This route is ECV-friendly.
By the way, yes, you can park at the Polynesian, or any other resort where you have a dining reservation.