Both umbrellas and coolers are permitted at the parks. There are, however, 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.
Unfortunately, there is no place near the rides to leave personal items. There are lockers available to rent near the front of each park if you'd like to store your belongings. You are welcome to bring things like soft coolers, purses, compact umbrellas, and backpacks onto the rides with you.
Have a great trip!