You should check out he disABILITIES! board. This is a regular question over there. You'll also find a bunch of dining reviews from people who are gluten free. My signature has a bunch of links to my dining reviews though we're off a heck of a lot more than just gluten and DD13 wasn't always off gluten in most of the earlier reviews.
WDW really is the best place you'll ever travel gluten free.
Personally I don't eat at many restaurants that aren't owned by Disney because they just don't all have the same kinds of standards and alternate products that you'll find at the Disney owned restaurants. The most notable exceptions are Teppan Edo, Raglan Road, and Kouzzina which get amazing reviews.
I've never had to figure out where to go to speak to a manager at a QS location. I typically go before the meal rush when the place is pretty empty. I ask first CM I see when we get in and they'll direct us to the right place. Sometimes we wait to the side while she gets a manager and sometimes we're handed the allergy book and told to get in line and the manager is called when we get to the front. Asking really does make it easy. If there's no CM anywhere near the door then we just get in line. The exception is Sunshine Seasons in the Land in Epcot where I just walk up to the station with the least number of people and ask to speak with a chef. With only a gluten issue depending on what you want you might not even have to speak with a chef (we've got 3 of us with lists of foods we have to avoid so our situation is much more complicated than yours). Last I heard, the entire grill station is gluten free.
If you want something other than burgers and chicken tenders at CS locations in the parks (the tenders really are amazing so you should try them at least once) then try these locations.
MK: Cosmic Ray's (rotisserie chicken, ribs)
Epcot: Sunshine Seasons (soooo many options like salmon, pork chop, rotisserie chicken, seared tuna, etc), Mexico (tacos on soft corn shells with rice but don't get the chips)
DHS: not sure because we always eat TS here
AK: Flame Tree BBQ (chicken, ribs, pulled pork sandwich) ** make sure to check out the allergy kiosk for more options and to get some good gluten free treats including stuff from Babycakes
DTD: Cooke's of Dublin (to die for fish & chips and onion rings), Babycakes (dedicated gluten free bakery)