No, there's no "black box" on the shuttle. What they do have is very much more information in telemetry data than any airliner would ever have. What they have is actually better than a normal black box on an airplane.
A "black box" is designed to survive a crash at hundreds of MPH, not thousands of MPH. A regular aircraft CVR or FDR would not have survived the catastrophic destruction of the shuttle. An airplane lawn-darting into the ground is one thing, but a shuttle returning from orbit at Mach 18 is a whole different problem.
Actually the shuttle didn't crash at thousands of MPH. It crashes at no greater speed than "an airplane lawn-darting into the ground." The terminal velocity of a falling body occurs during free fall when a falling body experiences zero acceleration. This is because of the retarding force known as air resistance. Air resistance exists because air molecules collide into a falling body creating an upward force opposite gravity. This upward force will eventually balance the falling body's weight. It will continue to fall at constant velocity known as the terminal velocity.
The shuttle was "slowing down" until it reached terminal velocity, by which time it had broken up and therefore was almost certainly falling slower than and "airplane lawn darting."
Then how did the videotape survive? It would seem that videotape would be even less likely to survive.