You can have an IT person from the hospital, from where the doctors in question work, set up a server with cloud software - essentially setting up your own cloud server with no 3rd party involvement while maintaining privacy issues within the said hospital. Whenever you are dealing with cloud computing, an IT person has do have access to the data/server with the data to perform upgrades, maintenance and security. So, to remove privacy concerns, this IT person must be associated with the hospital so that he/she complies with HIPAA guidelines.

WRT "...a customer to purchase their own cloud?" Ans: each patient under the care of these doctors would have to each setup and maintain their own cloud server.