Unlike many other theme parks, Walt Disney World does allow you to bring food into their parks. Here is the policy from the official Walt Disney World website:
Guests are allowed to bring food items—such as snacks or foods that do not require heating—into Disney theme parks. Inform a Security Cast Member of any food items when you enter the park.
There are great places at most of the parks to enjoy a bag lunch. We have frequently brought a small cooler containing frozen water, fruit, trail mix and granola bars into the parks. One great place to enjoy a sack lunch or dinner is Tom Sawyer Island at The Magic Kingdom. There are picnic areas where you can sit and eat and they are shaded so you are out of the heat of the sun.
There are places to enjoy your meal at each of the parks that are out of the way and shaded. You can also take your own food into any of the air conditioned counter service restaurants and eat there as well. Just keep in mind that if the parks are busy, counter service eateries are going to be busy as well.