The mathematical program with switching constraints (MPSC) is a kind of problems with disjunctive constraints. The existing convergence results cannot directly be applied to this kind of problem since the required constraint qualifications for ensuring the convergence are very likely to fail. In this paper, we apply the augmented Lagrangian method (ALM) to solve the MPSC and an application of the MPSC (i.e., the either-or-constrained program). We show that, under the MPSC relaxed constant positive linear dependent condition recently proposed in the literature, the feasible accumulation points of the iterates generated by the ALM are guaranteed to be strongly stationary if the multiplier sequence is bounded. When the multiplier sequence is unbounded, the feasible accumulation points are weakly stationary if MPSC linear independence constraint qualification holds. Some numerical experiments are conducted and compared with the recently proposed relaxation method. The numerical results demonstrate the effectiveness of the ALM and show that ALM can find better solutions than the relaxation method.