Every fuel tank is vented. You can't take fuel out of the tank for very long without putting something back in to replace the volume of space that the used up fuel is no longer displacing. If the tank wasn't vented, the negative pressure inside would build to the point that either fuel would stop flowing out of it, or the tank would collapse.
BMW 7-series used to do this in the 90s, the tank venting would fail and the tank would collapse. Train tanker cars have safety valves to prevent them from collapsing in the same way. What happens if that valve doesn't work? Something like this: YouTube - Implosion of a train tanker car - must see !
Cars have vent tubes that run back to the fuel system on the engine, so that no fuel fumes escape into the air. They also have a carbon filter on that vent tube, also for emissions purposes. So, yes, you can run with a non-vented cap, but you cannot run without fuel tank venting.