In this paper, we propose and analyze a mixed $H^1$-conforming finite element method for solving Maxwell's equations in terms of electric field and Lagrange multiplier, where the multiplier is introduced accounting for the divergence constraint. We mainly focus on the case that the physical domain is non-convex and its boundary includes reentrant corners or edges, which may lead the solution of Maxwell's equations to be a non-$H^1$ very weak function and thus causes many numerical difficulties. The proposed method is formulated in the stabilized form by adding an additional mesh-dependent stabilization term to the mixed variational formulation. A general framework of stability and error analysis is established. Specifically, some pairs of $H^1$-conforming finite element spaces such as $CP_1$-$P_1$, $CP_2$-$P_1$, and $P_2$-$P_1$ elements for electric field and multiplier are studied, and their stability and error bounds are also derived. Numerical experiments for source problems as well as eigenvalue problems on the $L$-shaped and cracked domains are presented to illustrate the high performance of the proposed
mixed $H^1$-conforming finite element method.