We describe and analyse a globally convergent algorithm to find a possible nonisolated zero of a piecewise smooth mapping over a polyhedral set, such formulation includes Karush-Kuhn-Tucker (KKT) systems, variational inequalities problems, and generalized Nash equilibrium problems. Our algorithm is based on a modification of the fast locally convergent Linear Programming (LP)-Newton method with a trust-region strategy for globalization that makes use of the natural merit function. The transition between the global to local convergence occurs naturally under mild assumption. Our local convergence analysis of the method is performed under a Hölder metric subregularity condition of the mapping defining the possibly nonsmooth equation and the Hölder continuity of its gradient mapping of the selection mapping. We present numerical results that show the feasibility of the approach.