CL-350 First Officer reported descent below crossing altitude due to autopilot mismanagement resulted in an ATC alert and a CFTT event.

Narrative

We were on a transition to the ILS XX approach from ZZZZZ. We were cleared the transition and the approach I set the final approach fix altitude of 8000 after verifying all the fixes and altitudes and checking with the PIC; VNAV was armed and we were descending.I was on number one radio talking to the FBO and when I glanced up; I saw we were at 12500 ft. and by the time I got the PIC's attention and we were correcting back to 12800 ft. I think we got as low as 12200 ft. As we were correcting the altitude deviation Salt Lake Center gave us a low altitude alert and I told him we were on our way back to 12800 ft. We had passed ZZZZZ and it should've leveled off at 12800 ft. Somewhere before ZZZZZ1; We hit some pretty good turbulence; and I know the Captain used the speed brakes and I think he went to vertical speed to Slow us down. After the speed was slowed down; I think he rolled the vertical speed back down to catch the path; but I do not believe the VNAV was on at that point. It may have been bumped and turned off. Unfortunately; I did not notice the VNAV being dissected. I was also distracted; trying to contact the FBO in the mountainous terrain. I should've just been backing up the PIC. I might have noticed the VNAV being selected sooner. Suggestions: When you are descending don't be afraid to slow the aircraft down when you see conditions for turbulence coming. Don't wait until the last minute. That was the distraction that I think caused us our deviation.