Of course they do. When I worked Psych in the NAVY I figure 1/3 of the Doctors were gay. There was no secret to it either, no one was in the closet and this was Portsmouth Naval Hospital which has a very large Psych unit. We had quite a few Doctors. Even one of the senior Corpsmen was a professional drag queen. Came to work with his makeup some days.
I certainly have known some gay shrinks, but 1/3 is more than my estimate; perhaps the ones I know are successful at hiding in the closet.