My sister is a vegan and has become an expert in Disney dining lol.
Counter service - There are sandwiches that can be made vegan at Columbia Harbour House in MK and Backlot Studies in DHS. It will take a little longer because they will have to make it fresh to leave off the cheese or sauce.
Sunshine Season in Epcot has good vegan options.
Animal Kingdom doesn't have good vegan CS options, or at least something that appeals to my sister. She loves, loves, loves Tusker House in AK so we do tableservice there.
There are always veggie burgers or simple salads but there are also different CS options if you plan well.
Wolfgang Puck Express in DTD has some good options.
Table service - You can special order an entree made up from any ingredients you see listed on the menu or have ingredients left out of a dish. You can also call a week ahead and speak with a chef from whatever restaurants you have ADRs, let them know when you are coming and that you are a vegan.
If they do not normally have tofu on hand at that restaurant, they can order some for that night and make you an entree not on the menu. My sister even has her own tofu entree made at buffets, and has one of the chefs walk her through the buffet pointing out the vegan options. Something may appear vegan, but as you well know may have something like chicken broth used in it that you can't tell just by looking at it.
Have fun!!!!!!!!!!!