All of the table service restaurants have GF rolls. Be Our Guest has Erin McKenna breads for breakfast in lieu of the pastry platter. The QS places have GF buns and breads for sandwiches.
Sanaa’s bread service with dips can be made with GF/DF naan bread - definitely go for all 9 dips, lots of variety
Jiko's Taste of Africa bread service can be served with GF pappadum
Tiffins serves complimentary GF pappadum with pomegranate oil; can also do bread service with dips there & at Nomad Lounge with the pappadum (Nomad Lounge also has GF churros!)
Hollywood Brown Derby Lounge serves their charcuterie board with GF garlic toast slices which are amazing
As previous poster mentioned, almost all TS restaurants have GF rolls, served with a variety of delicious butter/spreads: Jiko's fennel butter is awesome, Yachtsman serves a whole head of roasted garlic and salted butter with their rolls, Le Cellier serves a maple sea salt butter, Mama Melrose serves a delicious herb and spice olive oil
Raglan Road, the Boathouse, and D-Luxe Burger at DS all have delicious and fluffy GF burger buns (Udi's?)
QS restaurants are extremely accommodating of food allergies and will have GF breads, buns, pizza crusts, pasta etc.
Erin McKenna's at DS is GF/DF dessert mecca - the donuts are my favorite, but they also sell breads, cookies, pastries, etc.
I haven't been yet, but I read that Sassagoula Floatworks at Port Orleans has GF beignets - will have to check those out next trip!
They don't always know how to treat the brand of rolls that they have available though. Some restaurants will gently warm the rolls, which they NEED to do for them to be edible frankly, while some don't. The difference in quality of the experience is therefore massive restaurant to restaurant unless you make a point of requesting that the roll be heated (in a way not open to cross contamination).