I have flown air ambulance before for several years. Of course you need authorization in your FAA Operating Specifications to run or even advertise as an air ambulance. Pilots and mechanics really dont need any formal training to fly air ambulance but the medical staff will, especially if they are going to be called "crewmembers" and give preflight briefings and all that good stuff. Along with your onboard medical crew, you will need a Medical Director that will oversee the operation and be responsible for everything that happens, its his/her name that will be on all the records. There is alot to an air ambulance.. good luck.
