At the four major parks (AK, EC, MGM, MK) coolers are restricted to "small, soft-sided, which can fit in a locker. At the water parks any type or size cooler is acceptable.
The only restrictions are nothing in glass containers and no alcoholic beverages. I have seen people come through the gate hauling wheeled coolers about 4' x 3' x 2½'. Note that the cooler can be on wheels or carried, but the way the rules are written you can not have it in a wagon.
Some things people don't think about (which tend to be in glass) are pickles, salsa, Mayonaisse, Starbucks drinks, Snapple drinks, Tabasco.
If you show up at the gate with prohibited items we will give you three choices:
(1) Take prohibited items back to your car or resort.
(2) Put prohibited items in the nearby trash can.
(3) (Only for other than alcoholic beverages) We can give you some 16oz paper cups (or if you have plastic bags) you can dump the items out of the glass containers and throw away the container.