Hi, Annie! There are a lot of rooms at the Walt Disney World resort, and if you don't mind where you stay, you may not have any problems booking a room at the last minute.
I will caution you, however, that you are booking during the free dining promotion. That has a tendency to bring people down to Walt Disney World at a time when they otherwise may not have considered vacationing. If you are hoping for a very specific type of accommodation, you may run into some trouble finding availability.
If you can be flexible about your resort, you may be able to find something. I would say, though, that if you are going to be looking for a certain type of accommodation, you should do what you can to book as early as possible.
If you have not looked into the free dining promotion, you should see if you might be able to book that option for your family. It may save you money in the long run and help you in booking your whole vacation now.
I hope you have a wonderful time!