We used the CentraCare near Downtown Disney for my DD's asthma one time, but she was treated in the First Aid center first. They offered to call an ambulance, but at that point I felt she was stable enough for me to drive her. However, they did not take our insurance so I had to pay up front and then get reimbursed at home (I guess they take most major insurance companies, though). Also, they share the building with Turner Drugs, but Turner does not take insurance at all. I would think for an emergency like you are describing, your best bet would be to go through the park paramedics, especially because they will give you free transport to the ER if need be. We found them to be very helpful and quick to respond.
Hope all goes smoothly on your trip,
Mary