This is acutally my biggest concern with our upcoming trip. I have four vegan kids and the number of vegan offerings that are spicy are too many....
@mum2simon: Please check out
Vegan Disney World for lots of into but be careful when reading the articles there, since some of them are not recent.
Here are some not-at-all-spicy vegan foods that I regularly eat at WDW. And that I believe are still available, since I was just there last month. Some of them are off-menu items.
MK TS: Tony's: spaghetti with vegan meatballs (not spicy)
MK TS: Skipper Canteen: falafel (it's an appetizer but there's plenty for a meal); steamed veggies (at least broccoli but usually something else)
MK snacks: popcorn and Mickey pretzels are both vegan--this is true throughout WDW
There are probably fake-meat burgers at some QS places in MK but I don't like Impossible or Beyond, so I'm not familiar with these. There's a vegan entree at CHH, but I haven't had it and their last vegan entree was awful (the Doom Burger) so I can't vouch for the new one, a vegan "crab" cake.
MK area: GFC: On-menu is avocado toast. Also you can get vegan Mickey waffles there (specify when ordering). Off-menu is miso tofu with green beans, which is just the tofu version of the miso salmon. You have to ask your server for this. There's usually a vegan dessert there as well, but I'm not sure what it currently is.
DAK TS: Y&Y off-menu is sweet and sour tofu, which is the tofu version of the chicken. You have to ask for it. It's quite good. Comes with delicious rice. They also once made me a bamboo steamer full of steamed veggies. Definitely not on the menu. They have other off-menu vegan items there as well, but you have to ask, and I would stay away from the vegan version of the veggie lo mein, as it's pretty terrible.
DAK QS: Pizzafari (be careful, they close early--check the time) has Grains and Greens salad, which is very tasty and filling. Right on the menu and you can mobile order it. They also will make you a vegan pizza on request. It's okay, but Blaze it ain't.
DAK QS: Dino-Bites has Tofutti and the waffle cone is vegan. They usually have both vanilla and chocolate Tofutti. You have to ask for it. It's not on the menu.
I've never had anything from Satu'li since I believe the vegan food there is spicy.
DHS TS: Mama Melrose off-menu: spaghetti marinara. Not at all spicy. Make sure when you order it to specify no cheese, since they'll just sprinkle Parmesan on top of everything if you don't say you don't want it. Also, the bread service there is vegan and comes with olive oil, which you might want to ask for w/o the herbs, since they're a bit much. When I was there last month, they had vegan chocolate cake for dessert with Tofutti a la mode if desired, and it was delish. I think they have some on-menu items there as well. Take a look.
DHS QS: Woody's Lunchbox on-menu: vegan (or, as everyone says now, plant-based) grilled cheese sandwich with potato barrels. Delish and fun to eat. They also have vegan tchotchos, but maybe they're a little bit spicy. So perhaps not.
Epcot TS: R&C (dinner only these days, unfortunately): off-menu vegan fish & chips with vegan tartar sauce and French fries. Tasty! And some on-menu vegan items as well, maybe shepherd's pie? I've never had it so I can't vouch for it.
Epcot TS: 9 Dragons (China): on-menu (but 9 Dragons doesn't list what's plant-based--you just have to know): spring rolls (in the restaurant only--the QS place has no vegan items) but the sauce is not vegan, however, the spring rolls are great, no need for the sauce; and the tofu and veggie entree is also vegan as is the rice.
Epcot QS: Katsura Grill (in Japan): on-menu vegetarian sushi. This has ginger in it, which I take out, but many people don't consider ginger spicy, so that's up to you.
Epcot QS: Sunshine Seasons has various vegan items available at different times and it's constantly changing, so I don't want to tell you about something that might not be there when you're there, but they almost always have Enjoy Life chocolate bars around (kind of hidden--just ask) and usually a hummus entree, but the latest one was a disappointment. And the chocolate mousse is vegan. And of course Oreos are unintentionally vegan.
I hear that the vegan food choices at the Germany TS restaurant are terrific, but I've never eaten there. It might be something you want to check out.
That's all I can think of at the moment.