I’ve driven from from El Salvador to Washington State in a VW camper van, camping all the way and had no problems (except a few mechanical ones—but pretty much every mechanic in Central America and Mexico is a VW expert).
To get all the way to South America is a little different. Passing from Panama into Colombia requires passing through the dense jungle in Northern Colombia. This area is controlled by the FARC and/or the ELN (FARC=largest guerilla army, ELN=another substantial guerilla army). Both use international hostages as leverage, and thus it is not safe to travel through this part of Colombia. Anyway, you will probably get stuck in the mud on some dirt roads in the jungle, if they even go through. Don’t try it.
In Panama you can drive your car onto a barge/ship and skip this northern part of Colombia and enter further south where the FARC is not in control. Not all of Colombia should be skipped in my opinion—it is an absolutely stunning countryside, with beautiful colonial towns and cities, and some very cosmopolitan, modern cities. Do your research on what parts are safe to travel in. Don’t just use US state department recommendations—research a little further.
One can also skip Colombia altogether and disembark further south in Ecuador or Peru. From there you will definitely hit some windy mountain roads, but should be able to cruise all the way to the southern cone.