I would imagine that as long as you've got your doctors orders, Celebration Health could draw the blood. I'd call them before you leave to make an appointment with thier lab (if they take them) and check on insurance info.
There is an urgent care center not far from the main gate of Disney. They take nearly all insurances. They will also pick you up and return you to your hotel. I had to take my son there in June. Unfortunately I can not remember the name of the center but I was told about them by Disney. So you might try calling Disney and finding out their name and them contact them about blood work.
my father has to get his levels checked all the time and he travels for business. he has a prescription and he just finds the nearest place and they do the work and call the dr.