From the Universal Website:
Acceptable items:
•Bottled water
•Small snacks that do not require heating
•Any food required for medical purposes and medically–indicated nutritional supplements
•Any food required for special dietary needs
•Baby food/baby formula
•Soft-sided insulated bags no larger than 8.5" wide x 6" high x 6" deep
Prohibited items:
•Picnic lunches
•Food that requires heating or refrigeration
•Alcohol and glass containers
•Hard-sided coolers
•Soft-sided coolers larger than 8.5" wide x 6" high x 6" deep
•Coolers, suitcases, and bags with wheels
Please be advised that all bags and personal items are subject to inspection.
They do open every bag and look through it, usually with a small stick of some kind (one was even using a wand from WWoHP!

) I brought a camera bag and another bag, both were opened and inspected. Mine were inspected at the main entrance to City Walk, I did not see them doing inspections inside US or IOA.
But as long as you don't have anything on the prohibited list, you should be fine, and it doesn't take long.
A small cooler with a few frozen juice boxes (or other drinks) and snacks should be fine. I usually always freeze the drinks overnight beforehand, then I don't have to put an ice pack in the cooler.
For $8-$10 per day, you can rent a locker in either park that you can access as many times as you want, if you need a place to store it while you're on the rides.
ETA: The only place I ate at IOA was Three Broomsticks. And while I thought the food was good (considering it was prepared in a theme park), and the adult sized portions were okay, the child portions were very small - more like a "preschool" portion - and the kids at the table next to me did not appear to like their food. If you plan to let the kids snack all day at various places, it's probably an appropriate portion, but if you want to just have one good-sized meal and maybe a snack or two later, I'd choose someplace else.
There should be a sticky in this forum that shows all of the menus for the restaurants, if you want to see beforehand what it might cost in some of of the restaraunts.