A good place to start is with a hospital or hospital system, if you have one nearby . Most of them have their own volunteer coordinator or a volunteer coordination office that will place you in a clinic.
I am currently volunteering through two hospitals. Both of them had volunteer coordination offices that I contacted directly (found contact info via the hospitals website). I had to fill out an application to volunteer, and then they matched me with a placement based on my availability and what type of setting I wanted to be in. I also have 3 opportunities per year to be reassigned to another clinic, making it easy to see a variety of patients. They really make it very, very easy for you.
And as far as supervisors are concerned, it varies at each place. One of my placements has 2 coordinators and at least a dozen therapists each time I'm there. The other has one therapist and one aide, and that's it. Since I'm going through the volunteer office, there is a person there that is considered the supervisor of all the volunteers.