Since the M&P is not in use in any great numbers by our military, and since it is not officially adopted, our military handguns being the M9 Beretta and the M11 SIG Sauer, I doubt that many military persons have taken the M&P Armorers course. Therefore, by definition, nearly everyone who takes the M&P Armorers course IS a civilian. Police in the United States are civilians, unless on active duty in the military.
The question, therefore, is whether S&W will allow a person who is not in law enforcement to take the course. I know that some dealers and distributors have done so, as the S&W web site lists dealers with M&P Armorers on staff.
If you are willing to pay the fee and travel to the course, and if seats are available, I cannot see why they would not allow it, as they are not likely giving out "state secrets" or "nuclear launch codes," however, strange rules often accompany training of any kind associated with law enforcement as sworn personnel seem uncomfortable in such settings around non-sworn personnel, at least in my experience in taking the Glock Armorers course six or seven times over the years since 1985. Perhaps you could call the factory and ask.