D-Horse is your first buddy, usable from the first mission in Afghanistan.
Your bond with D-Horse will increase as you ride him, by 1% for every 36 seconds of riding. That’s a solid hour’s worth of riding around on his back, so it is a bit of a grind.
Riding during Side-Ops and missions obviously works. I’ve had mixed results with this in Free Roam mode, I’m thinking the time will count if you’ve set up a mission or side op-marker (to commence from the map rather than ACC) on your map, rather than aimlessly wandering.
Hero’s way (Ep 3) is a good mission to get this on, lots of open space to the north of the village with few enemies, no time limit on enemy movement, so you can land to the north, ride around for ages, then snipe the commander and ride back out again. Rinse and repeat.
NOTE: if you hit the 100% you’ll either need to exfil from a mission, or start one and then exfil, for the trophy to register, it will only pop on a results screen.