You can not travel the speed of light, you can only get close to it and the closer you get the more relativistic effects you experience, but we can continue talking about the problem you posed in the context of approaching the speed of light.
From your perspective
As you approach the speed of light, the mirror actually gets closer from your perspective and the speed of light both approaching you from the mirror and leaving you moving toward the mirror is $c$. However, you still perceive the mirror moving toward you. So if the mirror is moving toward you at $0.99 c$ and the light from it is coming at you at $c$, then the light from the surface of the mirror only departs at a relative speed of $0.01 c$, which for the context of this discussion we'll say is not very fast.
From the mirror's perspective
The mirror is a part of an inertial reference frame that "observes" (which is really just a formalism for relativity, different than seeing with photons obviously) you moving toward it at $0.99 c$ and let's also note that you are length-contracted like a pancake and experience time more slowly according to the mirror. Light comes from the spaceship (I presume) at $c$ and moves toward the spaceship at $c$. The light that the spaceship emits only moves $0.01 c$ faster than the speed of the spaceship.
In both cases it is agreed that the light from the spaceship emitted at a time $t$ before collision with the mirror only hits the mirror a small amount of time before the spaceship itself slams into the mirror (specifically $0.01 c t$, and yes I know this isn't an objective time measure as I have used it here). In the limit of going exactly the speed of light, then of course, no light from the spaceship is able to reach the mirror before the collision. A reflection of the spaceship in-travel is impossible in this case.
So what do you see?
You see reflections in the mirror from before you started your speed-of-light trip. If we talk about going close to the speed of light, like $0.99 c$ then you see a highly blue-shifted version of yourself for a short period of time (relative to the duration of the trip) right before you crash into the mirror.