Hi, Jennifer! There are a couple of things all working together here that may bring you news you don't especially want to hear. To start with, going to Walt Disney World during a free dining period is going to make the table service restaurants even more popular than they normally would be. Secondly, the popularity of some of the table service dining experiences is so great, that it's typically difficult to get a reservation for those restaurants unless you book in advance even when there is not an offer of free dining.
As a family of five, I will also say it can be tough to snag a last minute table with a family of this size, compared to grabbing a table for a party of two. We've had hit and miss luck getting last minute and walk up space at some of our favorite restaurants.
If you don't mind where you eat, you should be able to find something, somewhere. I recommend heading to the restaurants at off peak times - either an early lunch or a late dinner, for example. This may increase your chances of finding availability.
I also suggest that you consider making at least a couple of dining reservations - not so many that you feel locked in, but at least a couple so you know you are going to be able to get a table at restaurants you really want to try! There is no penalty for cancelling at most of the Walt Disney World restaurants, so if your plans change, you can cancel at the last minute. That might make someone else's day!
Have a great vacation!