Hi Julia:
One concern for my family at generic amusement parks is the lack of healthy food options. Disney, however, has made a huge effort to provide healthy food choices, and Guests vacationing at Disney World now have a lot of control over the food they eat. Even eating at a Disney counter-service location doesn't mean your family has to eat unhealthily. In addition to traditional park fare, we've found that counter-service locations have added more soups, salads, wraps, and vegetarian options. Kids can also order health foods such as baby carrots, fresh fruit, and applesauce as side items. At any Disney dining establishment, even at a counter-service restaurant, do not be afraid to make a special dietary request. If Disney is able to honor your request, they will be happy to do so. Also, every Disney Park has at least one fruit stand where you can purchase fresh fruit. Check your Park map for the location.
The variety of Disney menus and the lengths they go to accommodate dietary needs means you probably can find choices at all Disney restaurants (including steakhouses). But for specific recommendations, I would look at Sunshine Seasons for counter-service in Epcot (fresh made-to-order sandwiches, noodles, fish, soups and salads,For counter-service) and Columbia Harbor House in Magic Kingdom. For sit-down, I would look at Be Our Guest and Coral Reef, though you probably can't go wrong at any restaurant.
For my family, we always made advance dining reservations for sit-down meals to ensure we can eat at our preferred restaurants at preferred times. If you are traveling during a peak time (such as when schools are out of session), I strongly recommend making reservations. Otherwise, if you don't mind waiting or can risk not being able to eat at a preferred restaurant, then you may be able to risk not making an advance reservation.
Enjoy your vacation.