Do most hospitals and clinics require you to get a hepatitis b vaccine?

Employees=yes. There are such infection control requirements for employees in a hospital or clinic setting. This often includes not only hep B, but vaccines for the full range of potentially contagious diseases. Annual flu vaccines are also a common requirement. This is commonly set to prevent spread of contagious disease to patients and limit liability to the hospital if the employee is exposed in their work.
Hep B vaccine. Not if you do not work there. Only employees who work in hospitals and clinics are required to take the vaccine to prevent them from getting it from an infected patient.