E.B.
Yes, ours did, at age 2. It was an extremely helpful experience. Not only did he have insight into our dd's behaviors but he was also able to help us as parents. He explained that since she had medical problems, and some resulting behavior issues and psychological issues (fears related to food due to a gastro problem, for example), that we would have to have new strategies for helping her and dealing with her (different from how we dealt with her healthy older brother). His tips and advice and suggestions and insight were really good. We were able to implement his plans and we all made progress. Plus he helped us see that her difficulties and behaviors were logical and not the result of us being incompetent parents (which we were at the point of thinking).
We asked her medical specialist what kind of psychologist we needed. She was able to recommend the perfect person, because she understood dd's medical issues.
So if you already have a pediatrician for him, you might ask that doctor what kind of mental health professional would be best for your child. A behaviorist? A psychiatrist? An occupational therapist? There are so many choices, and a good medical doctor often has insight into what kind of mental health professional would be a good complement to his treatment.
Our dd did not see a psychiatrist who could prescribe medication, but a therapist who dealt with behavior issues.
Check your state's licensing board when you have some names and read up on their accreditations, licenses, any past complaints, etc.