You are certainly welcome to use the baby care centers for your son. In fact, during the brief time while I was making the video about the baby care centers, found
HERE, there were two special needs families using the facilities. The cast members that work at the baby care centers were incredibly sweet and helpful to them. The baby care centers can be a great oasis for a special needs family that needs a bit of privacy or extra help.
While I have not verified that absolutely every restroom at WDW has a changing table, the vast majority of them do, including the women's, men's, family, and companion restrooms.
I hope you and your son have a wonderful time at WDW!