A quick glance through the menus here on WDWMagic show an 8 piece chicken breast nugget meal is available at Cosmic Ray's Starlight Cafe, Columbia Harbor House, and Pincocchio Village Haus, for $7.69. A 6 piece chicken breast nugget meal is available at the Golden Oak Outpost (in Frontierland) for $6.89.
If you only want the chicken nuggets, most eateries at Walt Disney World will allow you to order them a la carte, which will usually shave off a dollar or two from the meal price.