I would avoid really focusing on this trophy until at least shanty town, the reason is, small animals are hard to hit when they run, but at that point you'll have the shot gun, you can then just shoot in their direction and kill them.
Small animals are all around the game. You'll be able to get chickens in a few places (mountain village). Rabbits are in the first game area (where you tracked the deer) and rats are in shanty town as well as other places.