Hi, Amanda! You can absolutely bring sandwiches and snacks into the theme parks with you. Generally, a smaller, soft sided cooler is acceptable, but nothing that you have to roll behind you, for the safety of other guests.
We often do a quick breakfast in our resort room, consisting of cereal bars, granola bars, bananas, and if we have a refrigerator, some yogurt. You can also use the coffee pot to make hot water for packets of oatmeal or Cream of Wheat hot cereal.
Use lunch as your bigger meal of the day if you are doing table service restaurants. The menus are often times very similar, but a bit less expensive at lunch time. The buffet restaurants are a great way to do a big meal so that you may only have to eat a light snack or small dinner in the evening.
Our family also likes to do food we can share. Two of my kids can share the giant turkey legs (and that includes having me take a bite or two!), and if they are looking for a "side dish" to go with it, I'll pull out raisins or chips from my backpack. I also sometimes order one adult burger meal from a counter service location, and split the burger and fries between two of us - one adult and one child.
You can bring your own single serving drink packets to turn water into iced tea or lemonade, and you can refill your water bottles at the fountains. The counter service locations will also give you a cup of ice water at no charge.
Have a wonderful vacation!