Check on the websites for specifics.
Many people had coolers at Aquatica. It should say on their website what the size allowed is (I read the signs, but don't remember - our "cooler" was a soft lunch bag style that we kept waters in). They do not allow sandwiches and family size items (huge bags or chips, 2 liter bottle of soda, etc.). You are supposed to go outside the park to have a "picnic." That said, I did not see them removing any items during bag check and saw plenty of people eating home (or hotel) made sanwiches with their big bags of chips . We had a can of pringles one day and they let that in without batting an eye. We had lots of water bottles.
I also saw some rolling collers (kind of like backpacks) at SW.