Hi, Carrie! My short answer would be that I hope so. The technical answer is that if any of the theme parks fill to capacity, they cannot allow any additional guests to enter the theme park. The Christmas holidays are a super busy time at Walt Disney World, and there are instances when a theme park does fill to capacity and has to be closed to additional guests, but I can't predict if your family would be affected.
In my experience, I have been able to enjoy dinner at one of the resort restaurants, and attend extra magic hours at the theme parks - even on Christmas Eve and Christmas day. But it is going to depend on the crowd level, and where the bulk of the traffic chooses to go.
I will also share with you that in my experience, Disney's Hollywood Studios has been a good bet for my family during busy times. It seems that the Magic Kingdom is especially popular on Christmas Eve and Christmas Day, and while we have been unable to gain entrance to EPCOT and Animal Kingdom in the past, we've always been fortunate enough to get into Hollywood Studios.
Hopefully, this will be your experience as well!
Have a wonderful vacation!