Which kind of doctor sees sensory processing disorder?

Team approach. The referrals usually begin with the primary care physician to Psychologists or Occupational therapists for evaluations and a determination of a sensory profile. Then treatment begins to learn modifications, accommodations, techniques and work-arounds for the sensory issues.
See neurologist. Although people use the term "sensory processing disorder" in reference to many different kinds of problems, a good place to start would be a behavioral neurologist. They specialize in brain processing disorders, not just alzheimer's.