There are two different ways we could answer that question. One is that, even though we're currently going at the speed of light, it's a scientific fact that light has to travel at a fixed speed in all reference frames. So our reference frame would see our headlights shine out just like normal.
However, everyone else outside would see your vehicle traveling very fast - much faster than the speed of light! But to them, the speed of light is the same as yours, so it would look like the light from your headlights were going slower than your car! This is one of the many paradoxes that can arise in relativity.
Unfortunately we can't say much more than that, since this paradox can't be resolved. This is why scientists are pretty confident that we could never travel faster than the speed of light.
However, if we were to break one of the assumptions of special relativity, namely that light travels at the same speed for all observers in all reference frames, then we would also see our the light from our headlights trailing behind us!
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