You first get the bow on Texas day 1. You may get this on your first playthrough and it isn't very difficult to do so. You will get the bow about halfway through the game. After you complete the brief training to use it you will get hunts that require you to use the bow. Simply keep using it while you can and it will unlock either on this playthrough or your Gun Master playthrough.
Tip: When it comes to hunting the trophy animals that aren't too far away, try your best to sneak up closer to them and take them out using the bow. Some of the trophies will need to be dealt with in this manor, but not all. Aim for the neck as it's a quick kill.