You're moving very nearly the speed of light with respect to some things in our universe already! With respect to those protons moving around in the LHC at Cern, you are moving at 99.99999% the speed of light. So, can you see yourself in the mirror?
But as Drakkith said, you can't actually travel AT the speed of light.
Do you not know why it is called relativity? You have to say you are traveling near the speed of light relative to some other object. But the mirror you are holding in your hand is motionless relative to you. If you can see yourself in a mirror when you are standing still then you can see yourself in a mirror when you are moving at high speed relative to something else.