My nine year old son has reached the age where he is worried about the Kid's Menu being too small also.
Walt Disney World Resorts and Parks do allow guests to bring in snacks.
My family brings in granola bars to enjoy while we walk around the parks, and that way the kids don't get ravenous before the actual meal! The portions are very generous at both the Counter Service and Table Service restaurants which means that I always have food left on my plate. And my nine year old has been known to reach over all ninja style and steal a few bits of fruit from my plate!
So, while I do bring granola bars for the kids to enjoy as snacks, we haven't actually attempted to open one up in a restaurant. I would quietly ask the restaurant's hostess or your server if it would be okay in that particular eatery. The Cast Members are always willing to help and are there to make sure your Disney experience is Magical!
Thank you so much for visiting us here at Walt Disney World Moms Panel! I hope that you and your family have a fantastic time with nary a tummy rumble!