We consider tunneling in a system of two parallel two-dimensional layers of electrons in the presence of a strong perpendicular magnetic field. The tunneling current between the two layers is blockaded at low voltages due to strong correlations in each subsystem. We calculate the current-voltage characteristics by modeling the two subsystems as Wigner crystals, and find a current peak at a finite voltage. The peak is broadened by the excitation of collective modes of the Wigner crystals. Our results are in a good quantitative agreement with recent experiments.