I am not getting involved on the whole whether or not the GF or YC bakeries can deliver cakes. What I do know is that many places do not allow for cakes to be brought in fromt he outside. This is a huge liability issue for them.
I had a very formal Sweet Sixteen and a historic manor and a huge guest count. At that point, there was a cake that I had my heart set on. There was only one baker in the area that could do it. Unfortunately, the manor worked with only one baker- one not able to successfully complete the intricacies for the cake I'd wanted. That baker was the singular supplier of cakes for any event held at that location.
When it was explained to me as to why they only worked with this bakery it made perfect sense- the location and baker entered into a legal agreement. For example, the manor would have been the ones sued had they served contaminated cake, or if there was some type of problem with it. This baker was the only one willing to stand by and say "We'll take the fallout." I would doubt that a Disney restaurant would allow for a Publix cake, for example, to be served on their property.
When my Mom married last spring, we had a small reception at one of the nice hotels on the water in her area. We did bring in a cake from Publix- it was delicious and beatiful... but a PIA to carry and handle! She also had to sign a legal waiver form about bringing foor into the restaurant. This was in Florida, too.
I'm sorry that you may have found yourself in this situation. Being on government property AND Disney property could really put a damper on any plans. I would talk directly to the forces that be and find out exactly what you can and can't do. You may want to send an email so that you have it in writing.