Is it possible to switch from Military police to civilian police?

The short answer is no. I don't know of any department that will not require you to go either through their training program or one that is ran by their state. Most will give you points for being a Vet, but MP and Civilian Police don't really translate as well as many people believe.

I’ve never heard of a transfer program to an actual civilian police department. I know that a bunch offer veterans preference.

Regardless of military background Law Enforcement has training/certification standards. You need to be POST certified for your Region/State. Once you have that you are on your way. But you can't just walk up and expect a job because you were an MP.