Instructions
Complete the Abraham Reyes mission, "An Appointed Time." This will unlock the Edgar Ross mission strand, in which the main character, John Marston, is asked to ride "shotgun" in a Federal automobile during several missions.
Travel to Blackwater and talk to Edgar Ross and Archer Fordham. They will allow you to start the "Bear One Another's Burdens" mission, in which Marston must find a wounded man named "Nastas" and carry him to the Federal automobile, fending off attackers as he goes. This is one of only two missions in the game that feature a car, and will unlock the "Great Men Are Not Always Wise" mission.
Complete the "Great Men Are Not Always Wise" mission. This mission features a car that the villain drives, but Marston will not be able to interact with the car during this mission. This will unlock the "And You Will Know the Truth" mission, which is the second mission in this series in which the player can ride in a car.
Begin the "And You Will Know the Truth" mission. In this mission, Marston is tasked with manning a Maxim Gun mounted on the back of an armored car. Unfortunately, this car will eventually be overturned by an explosion at the end of the mission, and Marston will no longer be able to ride in any cars (barring any unforeseen changes in the expansions).