I'm almost positive you have to be able to carry the cooler--nothing on wheels.
(somewhere--probably here--I read that anything on wheels has to be pushed rather than pulled. Therefore, wagons are not allowed--kids must be in strollers)
If you must have a cooler, bring one small enough to put in a locker. You can put a bottle or two in a backpack.
If it were me, I would get water bottle holders at home or WDW, and refill bottles through the day. Bring Kool-Aid, instant tea, etc, to make Orlando tap water drinkable.
PS--you know you can't bring in alcohol, right??