Can an object traveling at the speed of light have something within it also moving at the speed of light?

Sadly, objects with m (ANY m) cannot travel AT the speed of light, ever. Regardless of what you may have seen on Star Trek, relativistic speeds are forever beyond the reach of mankind. In theory you could go very close to the speed of light in your spaceship, but when you start doing the math you quickly realize that the amounts of energy needed to propel even a modest spaceship to say .9 C are, shall we say, prohibitive at best.