Basically - your kids will pay like an adult at any of the buffet style restaurants, but they usually can order off the kid's menu at counter service or at the sit down restaurants.
That said, the kid's meals at WDW are often REALLY kid's meals - meaning that you can't order a cheeseburger - check the menus at wdwig.com - Deb has a great listing of all the menus on site, including kid's menus. Most places, counter service especially, will have a grilled cheese or a mini hot dog combo available for the kids.
At the sit-down restaurants, most servers are very accomodating to those tweens and teens that still have children's palates. My middle daughter - who is VERY tall - she's the biggest one in the pic below, at only 11 - is an extremely picky eater. When she was 11, I did order and pay for her meals as a child, although I got some funny looks, since her 13 year old sister looked younger, and ate as an adult.
When we went back for her 13th birthday - she tried ordering off the adult menus, but couldn't eat, as the food was seasoned or came with a sauce! On our most recent trip in Nov, also 13, she decided to revert back to the kid's menu at most sit-downs, very happily. She did often order an appetizer or dessert to insure that she had enough to eat, as the kid's portions sometimes were small - but usually not! WDW is famous for really giving you more than you need!
