Charcoal burning cars don't actually burn charcoal they burn the gasses produced by heated wood; the same gasses that burn in a campfire. A charcoal burner is a two part system: A closed chamber with chunks of wood in it, and a charcoal burner to heat the closed chamber and make the wood generate gasses by a process called pyrolysis. Pyrolysis is a fancy word for chemical decomposition of a substance by heat. The flammable gasses produced by pyrolysis are under pressure and are routed to a carburetor of sorts, mixed with air and burned in the combustion chamber.
This is not a real efficient system and is generally used only if there is no alternative. Getting ready for a trip requires loading the pyrolysis chamber with wood, starting a charcoal fire and then waiting until combustible gasses are being produced. Then, the engine is started and off you go -- slowly. As the pyrolysis proceeds, the chemical composition of the gasses evolved changes and so does the power output and the necessary fuel-air mixture. Eventually, the gasses in the heated wood are exhausted; all that is left is charcoal. Then, you open the closed chamber, remove the charcoal and put it in the burner, next you take some wood from the load you have to carry and put it into the closed chamber, wait till it starts producing combustible gasses and off you go again.
This is not exactly a get-in-the-car-and-go type of operation and it is also not something to which modern cars can be readily adapted. Besides the fuel injector issue, there are no running boards on which to put the burner assembly.
So, minus some details, that is how "charcoal" burning cars work (or sort of work).
Below is a further well explained enlightment on the workings of a