The achievement requires four different vehicles but not four distinct vehicles; that is, the achievement can be completed with multiple of the same vehicle class rather than demanding a different type for each driver. One also does not have complete the level or even the entire "tank run" in four vehicles; merely being in four will unlock it.
It can be earned as early as right after the first door Exuberant Witness opens for the Scorpion. In between there and the start of the tank run is one M820 Scorpion, two vacant Type-54 Ghosts, one vacant Ghost Ultra, and one vacant Wraith Ultra. There are also piloted Wraiths, Ghosts, Phaetons, and one Banshee which can be hijacked with the aid of a plasma pistol or the Spitfire variant, but this will be much harder to do.
The achievement is easier to earn on co-op for communication with one's teammates, but is still doable on single player. The challenge will be keeping your fireteam alive, as on Heroic they can easily get the vehicles destroyed. It thus helps to clear the area of enemies first with the Scorpion before ordering the other Spartans in their own vehicles. To keep them from getting on the vacant Ghosts just around the corner from the Scorpion, use the squad commands to order your fireteam to the right side of the tank before piloting it. Then stick to the right side of the road so they don't notice the Ghosts, clear the whole area leading to the Wraith, then head back for the Ghosts to order them in before returning to the tanks.