Current topology optimization of Reissner-Mindlin plates faces dual challenges: the inaccuracy of global stress aggregation in capturing peak stresses and the numerical instability caused by shear locking. This study proposes a rigorous framework that bypasses stress aggregation by enforcing local stress constraints directly via an Augmented Lagrangian (AL) method. To ensure physical fidelity across varying plate thicknesses, we introduce a locking-free polygonal finite element formulation. This approach constructs an assumed shear strain field along element edges, effectively eliminating locking phenomena without relying on reduced integration. The optimization scheme further integrates a vanishing constraint treatment to resolve singularity in low-density regions, with sensitivities computed efficiently via adjoint analysis. Numerical benchmarks demonstrate that the proposed method delivers superior accuracy in peak stress control and robust convergence for both thin and thick plates, offering a scalable solution for stress-critical engineering designs.