The "apparent" absense of gravity is explained (in classical mechanics) as being cancelled by a pseudoforce in the opposite direction, because you are inside a non inertial, accelerated, system of reference.
In order for a pendulum to oscilate, it requires two forces, one is the tension (a central force), such as the tension from the rope, and gravity, which keeps push the bob downwards. Now, in absence of gravity, there is nothing to push the bob downwards.
When you have a central and cosnatnt atractive force, the movement is circular.
In your example the testion will be zero if the bob is at rest (you either stop it or it was not oscillating). But if you give the bob an impulse, ther cord will be tensed, regardless of the absense of gravity. In particular, if you give the bob a motion that is perpendicular to the rope, it will undergo a uniform circular motion, because the cord will have a tension (and there is no gravity to make it go back and forth).