We propose a new dual semismooth Newton method for computing the orthogonal projection onto a given polyhedron. Classical semismooth Newton methods typically depend on subgradient regularity assumptions for achieving local superlinear or quadratic convergence. Our approach, however, marks a significant breakthrough by demonstrating that it is always possible to identify a point where the existence of a nonsingular generalized Jacobian is guaranteed, regardless of any regularity conditions. Furthermore, we explain this phenomenon and its relationship with the weak strict Robinson constraint qualification (W-SRCQ) from the perspective of variational analysis. Building on this theoretical advancement, we develop an inexact semismooth Newton method with superlinear convergence for solving the polyhedral projection problem.