Hiya Erica,
Welcome to planDisney! As a mom of two, I understand dining with littles with limited diets. The list of foods my three-year-old will eat is not very long, so we spend a lot of time navigating her palate and the rest of the group. It can definitely make things tricky, but I hope I can help you prepare for your upcoming trip.
I took a look at the menus for 
Garden Grill and 
Akershus Royal Banquet Hall (two of our favorites), and reached out to a Cast Member for confirmation. Unfortunately, neither place serves chicken tenders or nuggets at their locations. But don't worry yet —there are a few options. First, you can ask a Cast Member at either location if they can prepare chicken nuggets for your little one. There are no guarantees, but they may be able to provide some. Second, when I was doing some research, I found that some locations at Walt Disney World Resort allow you to bring in a quick-service meal for your little one, but that is up to each location's discretion. I would recommend heading to the dining location at least 3 hours before your reservation (you can cancel a reservation up to 2 hours before without penalty) and speak to the Cast Member at the podium if you can bring in a meal for your son during your meal. My last recommendation is that if you and your family don't have your hearts set on Akershus or Garden Grill as a character dining meal. If you all want a great meal that is also princess-forward, then I cannot say enough good things about 
1900 Park Fare. The food is delicious, and they do have chicken tenders at the buffet. I was pleasantly surprised to get my toddler to take multiple bites during our reservation, and we were able to be more adventurous. Another great option is 
Crystal Palace at Magic Kingdom Park if you want something inside a theme park!
I hope this was helpful, Erica, and you can find some great options for you and your family during your upcoming visit. As you continue planning your trip, please come back and see us if you need anything else. We would be happy to help you ensure that your vacation is practically perfect in every way.
Happy Planning,
Ashley F.