<<Disclaimer: Verify this information before applying it to your situation.>>

We just got back from our trip to WDW with our 5 year old celiac daughter.
 We ate at a number of the places people on this list have recommended, so
I won't repeat that information.  I will tell you about our experiences at
Ohana at the Polynesian Resort and the Coral Reef Restaurant at Epcot,
because I don't recall having seen those restaurants discussed here
before.

Ohana's dinner is served in two courses.  First are "starters" that
include fried wontons, chicken wings, slaw, stir fried veggies, fried rice
and a few other items.  Next come skewers of pork, beef, shrimp and
turkey.  We spoke to the chef on arrival.  Although the chef first said
all items except the obvious (fried wontons, bread, etc.) were GF, when I
quizzed him about soy sauce, which they use in a number of the menu items,
he checked and determined that they use Kikkoman, which is derived from
wheat.  But, he offered to make the fried rice and stir fried veggies
without the soy sauce, and he told us that the beef and pork were prepared
without soy sauce.  So my daughter had a pretty good meal topped off with
a generous bowl of vanilla ice cream.  And she really loved the hula
hooping and coconut races to which the children were summoned!

The Coral Reef children's menus actually had a baked fish offering
(unheard of in my experience).  My daughter wasn't thrilled at the notion
of baked fish, so they fried it in corn meal for her.  It was served with
mashed potatoes and corn.

FYI, we are careful to avoid obvious non-GF items when dining out, but I
would not say that we are scrupulously GF on those occasions (e.g., I
don't sweat cross-contamination in friers or mono and diglycerides).  So
take that into consideration if you are considering dining at these
restaurants.  I may not have asked all the questions that you would ask.
But, we ate out 5 days in a row and saw no adverse consequences.

All in all, the WDW chefs were always willing to discuss menu items,
answer questions, and made sure we were happy!