the wellness center where I teach some classes (while not yoga) and which is part of a hospital has yoga instructors as employees who teach pre-natal yoga as special wellness classes.
I would be surprised if such classes under a hospital logo would be run by independent contractors because hospitals usually need to control the way it is done very closely for liability reasons. This is not to say that other hospitals may not do it differently.
I do not know of any yoga teachers who rent space at a hospital. This does not mean it does not happen, just that I have not heard of it. Karin is correct. In a hospital setting there are liability issues, as well as billing issues. The hospital might make an arrangement with a group to run a program, or a wellness center, like the one she mentions, more than an individual coming in to run a class.
Of course there are also teachers who teach for research projects. I do know people who are involved in this model. Getting into this type of teaching is a little more challenging, and is easier to find if you already have contacts in that community. Though if your skills, experience, alliance status, and so on are really good you could probably make it happen with a bit of research.
Funny thing is I know a number of physicians who are also taking RYT certification to teach yoga. There is a real interest in using it therapeutically.
I run an adjunct program for a hospital. The hospital director is the Health Educator for our community. I am paid as a contractor by the hospital. All classes are contractors paid by the hospital via the community health education department. You want to connect with the hospital community health educator and present a proposal. Other hospitals may do it differently.