Our family has 2 carnivores, 1 vegetarian and 1 vegan. We eat at almost any restaurant on property. Most have something on the menu for veg lovers, and if they don't the chefs will make something for you. Most are happy to do so, we've only had a few so so interactions over the years.
Here's the list of restaurants we go to with no trouble for everyone to eat at.
Skippers Canteen, Be Our Guest
Tiffins, Tusker House
50s Primetime, Sci Fi Diner, Brown Derby
Biergarten, Marrakesh, Chefs De France, Le Cellier, Coral Reef, Garden Grill
Ale & Compass, Flying Fish, Kona, The Wave, Whispering Canyon, Boma, Sanaa
We've eaten at others too, but these are our most recent places with much success.
Honestly, no worries about this... I promise you both of you will be fed well.