I think if you focus on places where the type of cuisine naturally has at at least a good vegetarian presence, you are likely to get more vegan options.
Things like the african/asian themed restaurants like Boma, Sanaa, Tiffins, Skipper Canteen, Yak and Yeti, Nine Dragons, Tokyo Dining etc. Also I would think the signature restaurants would put together something quite nice. Sci Fi oddly enough has a couple of vegetarian options though now sure how good the the vegetarian shepherd's pie would be if made vegan but the burger sounds decent.
For quick service in MK Columbia Harbor house has a vegetarian sandwich that could be vegan if the bread is ok. In AK there are lots of choices in little stands like falafel and samosas. IMO AK is probably the easiest place to eat vegetarian/vegan. In Epcot, Sunshine Seasons has great outright vegan options (more than one) and Tangeirine cafe has a vegetarian platter that can probably easily be vegan if you're over in world showcase... the only one I'm not sure of is Hollywood Studios, but they are a bit of a qs wasteland anyway. I'm sure there is something somewhere.
I don't always feel like eating meat so those are just the places I've noticed interesting vegetarian options.