depends on the position you want to install the camera. If you want it facing you, there is a mount for the handle bar, and you can rotate the camera facing forward or backwards. I wouldn't do suction mount, due to the extreme vibration on the snowmobile, and jumps, etc. I didn't understand exactly if you want to mount the bow on the snowmobile, or the camera on the bow, so here are the 2 possible answers I have: For bow and arrows on the snowmobile:, depends on the model you have, and how can you fit it either on the handle bar, or on the cargo area in the back of your snowmobile. Maybe something on the style of the Kolpin Rhino Grip Xl Rack/Handlebar Mount Kolpin Rhino Grip Xl Rack/Handlebar Mount look up for similar mounts that can fit your rig. Now, if you want to mount the camera on your bow.... I think the best solution is a wrist mount you put on your left arm (if you are right handed). I don't like the idea of mounting something on the bow, since it's going to change the way the bow "resonates" and change the balance on it. With the wrist mount, you can get the camera facing forward and aligned with your arm and your line of sight... Good luck!
There is are adhesive mount(s) that you could stick on the bow. There are mounts for flat and convex surfaces. The suction cup could probably be used on the snow mobile. Then you could move it quickly to another mount location for a different perspective.
The motorsports edition comes with a lot of good options, the surface mounts and different arms worked well on my streetbike. I bought the handlebar mount and used it to mount it the stabilizer on my bow
I have a head band mount and it would work on your snow mobile and for bow hunting. I also have a helmet mount that would work if you wear a helmet while on your snow mobile. I use it on my scooter, a Piaggio MP400 and it has produced some interesting video particularly at 150 km/h on a curvy roads. I also have a handlebar mount which may work on your snow mobile. I have a GoPro underwater housing which I plan to try out scuba diving in Maui next February on vacation.