Unless you're planning to attend a lot of character meals and/or buffets, the dining plan won't really save you any money and might cost you money. I know that people like it for convenience, but for me, the numbers don't add up. You should look at some menus, make estimates, and see if the numbers add up for you. For me, they never ever have. Not even close.
As other posters have stated, there are entire sites devoted to vegan dining at WDW. This is a great site:
https://vegandisneyworld.com/ and there are vegetarian/vegan threads on the DIS, also with good info. Be careful, though, as menus change all the time, so always check the official WDW site as well.
That being said, some QS restaurants have vegan options that aren't on their menu. For example, Restaurantosaurus at DAK has a vegan veggie burger that's not on the menu. I had it just a couple of weeks ago, so I assume it's still there, but I did ask someone before waiting in line. And Pizzafari at DAK will make a vegan pizza for you.
FYI, all the TS restaurants will make something for you, even if there's nothing on the menu. Just tell your server you're vegan and the executive chef will come out to chat with you. The caveat here is that some places do this better than others.
The most vegan-friendly park is definitely DAK. TH is a vegan paradise, Y&Y will make several different dishes vegan, there're carts all over the place with vegan choices, and there're the two QS places I mentioned as well.
If you're stuck with the dining plan--perhaps it's tied to your reservation and there's no way out of it?--then you can still make it "pay," I guess, by doing a lot of buffets. Boma and TH are great vegan choices.
Warning: the vegan QS choices at MK are scant these days. The Lighthouse Sandwich at CHH--vegan without the slaw--an old fave, was discontinued a few months ago. Plan carefully.
And, you might like to know that both the popcorn and Mickey pretzels they sell in the parks are vegan.
Good luck figuring it all out! It's really pretty easy to eat vegan at WDW. I personally wouldn't do the dining plan, though.