As a Marine myself, I can tell you that "Marine Corp's" is completely wrong.
It is technically correct to use either:
If you look hard enough you will find examples of both, and typically one may be preferred. I have never seen either stated as incorrect; one is either more or less preferred than then other.
I personally would stay away from using "Marine Corps's" not only because of personal preference, but because you will get the occasional mispronunciation as "corpses", which, while it is often a Marine's primary mission to turn an enemy into a corpse, we do not like to hear our name mispronounced as such.