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. If you absolutely felt like to needed to bring
more into the park with you, you could bring your food in two small
coolers, carried in by different people. This would also be fine.
And of course, yes, you can leave a cooler in your car.
Have a great trip!