You can not carry in glass containers or bottles, and no alcohol. Otherwise anything else is fine. Just be prepared to open the cooler for security.
I have heard that only soft-sided coolers are allowed- but have never seen this in writing. Just for convenience I can't imagine lugging around a hard cooler anyway!
So rest easy, and bring your child's food in with you. I also agree that WDW is accommodating for those with allergies, but sometimes it's not worth taking a chance.
-Sarah