You are welcome to bring your own food and drinks with you into the WDW
theme parks. You are not allowed to carry in alcoholic beverages or
glass containers (other than baby food jars), but other than that, any
food items are fine. And yes, you can bring a cooler with you into the
parks, but there are some restrictions on the size permissable coolers.
The exact page of the WDW website that discusses coolers can be found
HERE, near the bottom of the page.
The website states that:
Suitcases, coolers or backpacks with or without wheels larger than 24"
(61 cm) long x 15" (38 cm) wide x 18" (46 cm) high are not permitted in the parks. (Coolers required
for medication may be stored in a locker or at Guest Relations.)
So, as long as your cooler is smaller than 24" X 15" X 18", you'll be
fine bringing it with you. Most soft-sided coolers that I've seen fit
within those dimensions.
Have a great trip!