I'm allergic to corn too and LOVE Disney for all the special care they take to make eating there an enjoyable experience. I see someone already provided a link to more information and you'll find lots and lots of 'what to do' answers about food allergies.
On my last trip in May '08 I ate out almost every day and didn't have any problems. We also had a kitchen so I could cook my own food.
Side note - my allergy is not just to food - i have to be careful about inhaling corn as well (like the smell of popcorn). If the child is sensitive you may want to ask about the cleaners used in the room. For example Febreeze contains corn.
The items not identified as "corn" can cause trouble. When filling out the paperwork Disney gives you to specifically identify the allergy, be sure to include those hidden items - like 'xanthum gum' or 'dextrose'.
Its difficult to ask about ingredients at the fast service locations - they're loud and you don't want to hold up lines, so get as much info as possible before your trip.
Common sense is your best friend - fresh fruit and vegies always seem like an ok choice but how were they cleaned? Some fruit/vegie washes (like Fit) include corn. Definitly have corn free snacks on hand - sometimes you just don't want to take a chance!
Good luck and if you're going 1/26 - 2/2 I'll see ya there!