My husband and I are vegans and our time in the parks was the most difficult for finding things to eat compared to the rest of Japan. Your group, being vegetarians, probably won't have it nearly as hard. I'd imagine you'd have a lot of pastry options if nothing else.
We stayed at the Hilton and took advantage of their breakfast buffet every morning. They had lots of stuff we could eat and there would be even more options for you. During the day, we survived off our own snacks that we brought and supplemented with french fries
which were one of the extremely few vegan items they sold in the parks. There is one restaurant in each park that offers allergen meals -- Eastside Café in TDL and Ristorante di Canaletto in TDS, assuming it hasn't changed since we were there. We could've gotten a vegan/vegetarian meal from those places had we been able to get on the priority seating lists, but we weren't. You might have better luck with that but, either way, don't stress over it. You won't starve to death if you don't get in. You might just spend the whole day eating junk food.
After the parks, we'd head back to the hotel room and eat from the food we kept in the fridge.
The rest of the country was easier to navigate with dietary restrictions. Here are my tips:
Dashi is the ingredient that's going to give you the most trouble. It's a fish sauce and it's in just about everything. If you don't eat fish, you'll want to ask about dashi before ordering.
Get the Happy Cow app. It'll use your location to point you to nearby veg/veg-friendly restaurants. These are mostly little mom and pop type places and their hours of operation can be unreliable, so don't wait until your already starving to start looking for a place to eat. You might get there and find the restaurant closed... four restaurants in a row.
Worst comes to worst, go to an Indian restaurant.
Somewhere on this forum is an incredibly useful set of dietary restriction translation cards that explain in Japanese what you cannot eat based on your type of allergy/diet. We printed a small stack of the "I'm a vegan" cards and took them with us. They were invaluable outside of the vegan restaurants. Instead of trying to explain our diet through a language barrier we'd just hand over the card and, a few minutes later, that person would hand us food we could eat.
Make use of the convenience stores. Seriously. On the first day of our trip we walked into a 7-11 (and then a Lawson's and then another couple of chains...) and, using the 'vegan card,' asked the employees for help. They were incredibly accommodating and went through the ingredient labels of many of the prepackaged foods for us. We bought everything they said was safe and took pictures of the labels before eating it. That way we could match the pictures to the items next time we went into a convenience store, which happened at least once a day. I developed a bit of an obsession with umeboshi onigiri.
I make it at home to this day.
T's Tan Tan in Tokyo Station right before you go down to the platform for the train that will take you to Disney, is a vegan/fish-free ramen restaurant. It's definitely worth taking the time to stop there at least once.
