Disney rooms come with small refrigerators, except at the Value Resorts. You can rent a fridge at the Value Resorts, and it may be worth the extra money to help get your dining budget under control. A fridge allows you to keep a variety of foods in your room, either purchased from one of the nearby supermarkets (if you have access to a car) or from an online grocery-delivery service such as GardenGrocer.
If you opt for the Value Resort but don't want to rent a fridge, bring a cooler and fill it with ice from the hotel's ice machine. Bear in mind that perishables can only be kept fresh so long in a cooler. Pack ziplock bags, to prevent water from melted ice getting into your food. You'll also need to empty the water from the cooler in the bathtub each night before replacing the ice.
Even if it's just cereal or bagels and cream cheese, eat breakfast in your hotel room. Not only will you get a better jump on the day as you won't have to wait in line for food, you'll feed your family for less than the price of a single breakfast inside one of the theme parks. Prepare snacks to bring to the parks.
If you like to rest in your room in the afternoon, try to time it so that you can have a second meal there, at least one some days. This makes your food-price savings add up.
The cost of water alone for a large family can rack up hundreds of dollars during a weeklong trip (and can still be considerable for small families). Instead, buy a couple of cases of bottled water and keep them cool in your fridge. Take one with you in the morning, and if you return in the afternoon, grab another bottle before you head out.
If your fridge has a freezer, freeze your water to help keep it cooler longer. Remember, it can be hot in Florida---so you may be drinking a lot of water. You can also refill your bottles from the many water fountains throughout the theme parks, and some counter-service dining locations will give you free cups of ice water.
The cheapest---and fastest---meals on Disney property are the fast-food-style counter-service locations that can be found in each theme park. While not inexpensive, they offer the cheapest way to eat full meals inside the Disney parks. The Flame Tree BBQ at Animal Kingdom, for example, offers good BBQ sandwiches and pleasant outdoor seating.
Even if you eat your meals in the parks, avoid signing up for dining plans. Once you commit to one, you commit to paying for meals even when you might find cheaper alternatives. You also commit to buying snacks that you might otherwise skip.