When many-particle (vortex) assemblies with disordered distribution are subjected to a periodic shear with a small amplitude , the particles gradually self-organize to avoid next collisions and transform into an organized configuration. We can detect it from the time-dependent voltage (average velocity) that increases towards a steady-state value. For small , the particles settle into a reversible state where all the particles return to their initial position after each shear cycle, while they reach an irreversible state for above a threshold . Here, we investigate the general phenomenon of a reversible-irreversible transition (RIT) using periodically driven vortices in a strip-shaped amorphous film with random pinning that causes local shear, as a function of . By measuring , we observe a critical behavior of RIT, not only on the irreversible side, but also on the reversible side of the transition, which is the first under random local shear. The relaxation time to reach either the reversible or irreversible state shows a power-law divergence at . The critical exponent is determined with higher accuracy and is, within errors, in agreement with the value expected for an absorbing phase transition in the two-dimensional directed-percolation universality class. As is decreased down to the intervortex spacing in the reversible regime, deviates downward from the power-law relation, reflecting the suppression of intervortex collisions. We also suggest the possibility of a narrow smectic-flow regime, which is predicted to intervene between fully reversible and irreversible flow.