The first rule of transportation in RPGs: it's available as necessary to move the plot, and stops when the GM needs it too. If they want transit to be fairly easily available, that isn’t a problem. However, given the background of the Anarchy Zones, Motorized transportation would probably be a major bottleneck in the setting. Many parts would be hard to acquire, and the fuel more problematic yet - "Mad Max" this is not. In turn, part of the establishing self-sufficiency themes would be setting up fuel production, and then modifying vehicles to run off your supply.
One would think that there would be a lot of stranded and forgotten vehicles lying around, ripe for the salvaging. The abundance is true, but the pickings are actually rather slim. Cars are actually far more perishable than commonly thought. Lubricating oil turns to sludge, for example. Moisture infiltration means the upholstery might be a mold covered disaster. Rubber tires also degrade - most mechanics say you probably should avoid using ones that have been in storage for more than five or six years. This last piece of advice should make it rather clear that making fuel will be the easy part of getting the world going again. Importing rubber - that is going to be difficult.
Power sources also have limited shelf-lives, and the setting is taking place five years after most up-keep stopped. Batteries have probably discharged or corroded from time, to say nothing of being burned out by EMP. Petroleum derived fuels do go bad over time - partially evaporating, oxidizing, and absorbing water - leading to lower performance and more particulates in the fuel system. Biodiesl goes bad after a number of months. Ethanol by nature is antiseptic and hydrophilic, so it won’t rot, but it will contaminate itself fairly quickly.
These are just the issues that come with time passing, but remember - there are disasters in the background of the zone. Peak Oil and assorted emergencies in the preceding twenty years means that natural oil fuel and lubricants are quite rare - and likely guarded by remaining military for use in high performance war machines. As The Event occurred, EMP shorted out components, looters damaged what was left, and panicked people seeing reanimates for the first time often abused or crashed vehicles as a result. Since then aliens have been trying to deny the enemy resources, raiders have been trying to corner the supply, and reanimates...
Reanimates are a bit of an oddity. One of the big arguments for the existence of a type 4 is how they treat vehicles. Often, they will simply smash the things to prevent the victims from escaping. Other times, they seem to intentionally leave vehicles in near pristine condition, serving a lure to the unwary.
As stated in the beginning - making vehicles available to players isn't a bad thing, if it moves the plot along. A more likely scenario, however, is that acquiring the vehicles and maintaining it drives the plot. Finding a machine untouched in an underground parking garage, finding a mirco-fac that can produce replacement tires, and getting a working bio-fuel reactor going to keep it running, while fending off those who want it for themselves - that is the crux of getting the machine restarted in the zone.