Being vegetarian/vegan in Tokyo is possible, but difficult if you don't speak the language. I am vegetarian, lived in Tokyo for 11 years, speak & read Japanese, but eventually decided not to worry too much about whether or not there might be bonito flakes in the stock used to make miso soup or dipping sauce for tempura or noodles - it is very hard to avoid completely.
Conveyor-belt sushi places are often a good choice, as it's fun for the children and there are lots of rolls stuffed with cucumber, pickles, natto etc; egg sushi may have been made with stock containing fish-flakes. We often used to eat at soba noodle restaurants - buckwheat noodles and vegetable tempura (but again, dipping sauce may contain traces of fish). Tempura restaurants also good. We often used the Tenya chain of budget tempura places, where you can get a big bowl of rice topped with vegetable tempura for Y590.
In convenience stores, quite a few of the rice balls (o-nigiri) are vegetarian/vegan (eg stuffed with pickled plum or vegetables) but the labels are usually only in Japanese.
A four-day ticket to the Disney resort sounds a bit excessive, in my opinion. I have been to Tokyo Disneyland and DisneySea many, many times, and always found one day was enough to do most of the park, so two days for both. I think Tokyo Disneyland is smaller than the US versions. If you are only in Japan for eight days, devoting half of that to Disney seems too much - why not try to see some more Japanese sights?
In any case, if you are going in December, and avoid weekends/public holidays, the parks won't be too crowded and you shouldn't have to queue too long for the rides. We have often been on quiet days where you can walk off a ride and straight back on again with no queuing. However, if you are there over Christmas/New Year itself, the parks will be a nightmare.
Oh, and speaking of nightmares, there is virtually nothing that vegetarians can eat in either of the Disney parks, apart from popcorn and doughnuts. I used to smuggle in picnics - luckily they don't check bags too thoroughly.