The way around both of these issues, is to plan where to eat, book table service (sit-down/waiter-service meals) in advance and know where you can find the best veggie counter service and snack options. This might sound like a lot of fuss and bother, but in my opinion, it's worth it.
When planning your food options, your first stop should be AllEars.Net. This website is amazing - it lists up-to-date menus, vegetarian dining strategies plus recommendations for vegetarian and vegan counter service (fast food) and table service dining locations.
There are various Disney Dining Plans available which can either save you money, or may be 'free' as part of your holiday package. We got the free, regular Dining Plan (DDP), which entitled us to one table service meal, one counter service meal and one snack each per day, which was more than enough food!
Veggie kids on the DDP
My kids are 10 and 12, so in Disney terms, that makes them adults! I wasn't impressed with paying adult prices for them, but with hindsight, I'm glad they were able to eat from the adults' menu at no extra cost, purely for the better selection this gave them - even though the portions were far too big for them most of the time. For lunch, the children generally shared a counter service meal.
If you have younger veggie children, I would suggest either paying extra to upgrade them as adults, eating mainly at buffet restaurants, or making sure in advance that the table service restaurants you want to eat at will be able to provide small portions from the adults' menu but at a the cost of a child's meal/DDP credit. If not, your children will be stuck with a very limited choice consisting of mainly pizza and macaroni cheese.
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