There may be faster ways to do this, but I found that in mission 5 (Over the Fence) this is pretty easy to do. If you select the landing zone to the east, there'll be four sheep (and/or goats; can't tell the difference, sorry) nearby.
Just be a bit careful; there are two guards on foot patrolling the area, plus another guard that arrives in a jeep. I'd advise using D-Dog if available, because he'll mark the enemies and animals as soon as you start walking.
Let the guy in the jeep pass by and neutralize the other two however you like. I chose to use a silenced sniper rifle with tranquilizing rounds, but you can use an assault rifle as well. Just remember not to use lethal weapons on the animals, of course...
There's no problem using an unsilenced gun either, as the outpost is further away and no one else will notice you.
Once you hold
close to the four animals to extract them, press
to pause the game and select Checkpoint. You don't need to trigger a checkpoint for actually extracting them, just reload the mission when you're done.
There is a fifth animal to the right, but don't go after it. The game will trigger a checkpoint and the other four won't be there anymore when you reload the mission.
With this method, you'll have to reload the mission twice.
Edit: GLASS GHOST 0 tried to use capture cages but it didn't count towards the achievement, so you have to use Fulton extraction on them; it works with smaller animals, such as birds, though (you can put smaller animals away by holding
close to them). GLASS GHOST 0 also points out there's a bird nearby, on the billboard to the right of the LZ.
I tried this with an alternate account and also asked a friend of mine to try it himself, and capture cages didn't work, too. Fulton extraction and grabbing did count.
Hope the solution is useful; feel free to suggest any changes, and if voting negative, please tell me why so I can improve it.