In this paper, we present a posteriori error estimates of the weak Galerkin finite element method for the steady-state Poisson-Nernst-Planck equations. The a posteriori error estimators for the electrostatic potential and ion concentrations are constructed. The reliability and efficiency of the estimators are verified by the upper and lower bounds of the energy norm of the error. The a posteriori error estimators are applied to the adaptive weak Galerkin algorithm for triangle, quadrilateral and polygonal meshes with hanging nodes. Finally, numerical results demonstrate the effectiveness of the adaptive algorithm guided by our constructed estimators.