To repair the car, you'll need two items: A gas canister and a car battery. There will be at least two of each of these items around the map, so they should not be very difficult to find.
Once you have one of the parts, you'll need to find a car. There are two cars on each map; the blue car seats four counselors and the yellow car seats only two. Once you've found your car, take the part to it and install it. The battery will go in the front of the car and the gas in the rear. Begin installing the part by holding A, which will initiate a short minigame where you need to press either LB or RB as the meter passes the highlighted areas. Failing to press the correct button will cause a delay, but also make the sweet spot larger. Characters with higher intelligence will have both a faster-moving meter and larger sweet spots.
You do not need to start the car for progress towards this achievement, only repair it.