Some places here in Ca you can use your Medic experience to get a Coroners Investigator gig. Some counties will allow Basic experience, but they're few and far between.
Outside of doing death investigations, I don't see any real reason for EMS to be doing, or be part of, any investigative team.
Ca does have state agencies (Department of Insurance and Department of Consumer Affairs) that investigate insurance fraud, prescription drug abuse, illegal practice of medicine, and patient abuse among other things, but they're part of their own law enforcement agency.
My dream is to be approached by one of these agencies, or the FBI, to go undercover as a secret agent EMT to document/uncover fraud and abuse. It would be my own version of the movie Departed except with crooked LA ambulance companies.