Hi, Britt! What a great way to celebrate a sixth birthday!
There is a concierge desk at each resort, and some guests who are staying at the Club Level of certain resorts will have access to the Club Level concierge. The Club Level concierge can assist with advanced planning, including helping you secure dining reservations, golf tee times, and other activities, but if they are not available, there isn't usually anything they can do.
What I recommend is that you keep trying, either with a phone call to (407) 939-3463 or by visiting
Disney World Dining Reservations. Guests cancel all the time, and you may luck into a reservation that you are hoping to get. You can also check at the podium when you are there to see if they have any last minute tables open.
And I have to just throw in there - this is her first time at the Magic Kingdom! There is going to be so much for her to see and do, she may not even have a thought about the things she doesn't get to do on this trip. I think this is a wonderful surprise, and I hope you get to make some of the most amazing memories ever during your visit!
Happy birthday to your lovely little princess!