Disney is GF heaven! Every restaurant will have an allergy menu and many can make adjustments to regular-menu meals to make them GF, i.e. subbing out a side, leaving off a sauce or garnish, etc.
All Disney-owned sit-down restaurants will serve you some kind of GF bread service and all QS and TS also have dedicated GF friers for chicken tenders, fries, etc. Some of our favorite dinners that have extensive GF options are Jiko, Narcoossee's, Cali Grille, Yachtsman, Tiffins, Skipper Canteen. (Jiko and Skipper Canteen have the most amazing GF desserts - the Kilimanjaro and Quicksand, respectively).
We also love some of the lounges - Nomad Lounge at AK has an almost 100% GF menu (the wagyu sliders on GF buns are awesome, as well as the GF churros!). Hollywood Brown Derby Lounge also has a great slider duo of duck confit and wagyu, available on GF buns. And their GF charcuterie board comes with delicious garlic toasts!
Disney Springs - Raglan Road is not Disney-owned, but is awesome if you like onion rings, fish and chips, etc. - they can make almost their entire menu GF and they have a dedicated GF frier. The Boathouse doesn't have a GF frier, but they do have Udi's GF burger buns. Deluxe burger has a dedicated frier and GF buns. Morimoto has a small GF menu (sadly, I think they eliminated their GF pad thai). Jock Lindsay's Hangar Bar usually has some GF apps as well.
We have never had cross-contamination or any issues eating at Disney - I hope you enjoy some delicious meals!