The higher your Rifle Skill or Bow Skill, the less shots it should theoretically take.
You will need to continue to land shots until it drops. Moose, unlike every other animal in the game that can survive a bullet or arrow, do not bleed out. Go with a Survival Bow or Rifle and try to land shots to its body or neck. The most effective way to hunt a Moose in The Long Dark is to find a safe place to shoot from. If you do see one, take it down because it could vanish at any point and not return for days. If you setup camp right where the Moose should spawn, you’re unlikely to ever see it. Ensure you are entering an interior (or leaving the region) and coming back to check. If they are present, one will eventually spawn, but don’t camp the area.
It’s important to note that just because a spawn point is listed above doesn’t mean you will get a Moose on your instance.