I can think of three ways to answer this:
It can't happen.
It really can't happen.
Okay, that's probably enough ;-) Since you say we don't need to consider special relativity, suppose that the universe actually obeys Galilean relativity. That's the technical term for the intuitive way to think about motion, where velocities are measured with respect to some absolute rest frame, and there's nothing special about the speed of light or any other speed. If that were the case, then yes, the light beam would never catch up to car A. The energy contained in the light would presumably pile up in the headlight where it was emitted at first, but afterwards perhaps it would spread out sideways, or would be reabsorbed by the headlight as heat. We don't really have a good answer, because that's not the way the universe works - in fact, there's a lot of physics, both experimental and theoretical, that has been done to prove that it can't work that way. No matter how you try to resolve the problem, at some point you will run into a contradiction.
The best thing you could probably do would be to draw a parallel to some sort of wave that travels with respect to some fixed reference frame, at a speed much less than that of light. Sound, for instance. Sound waves travel with a certain speed with respect to the air, which defines a single absolute reference frame, and their speed is much less than that of light, so there are no special relativistic effects to worry about. Your headlight scenario would then be roughly equivalent to an airplane traveling at the speed of sound. What happens in that case is that the airplane creates a sonic boom, a shock wave which results from the energy in the emitted sound waves piling up at the airplane and eventually being forced to spread out sideways. So one might guess that in your hypothetical situation, the headlights of car B would create a light shock wave that would spread out perpendicular to the direction of motion.
This actually can happen in certain physical situations, namely when something is traveling through a transparent material that slows down the speed of light. This means that light itself travels at a slower speed, but not that the "universal speed limit" is any different. The effect is called Cherenkov radiation and it does indeed work out much like a sonic boom would.