A special finite volume element method based on postprocessing technique is proposed to solve the anisotropic diffusion problem on arbitrary convex polygonal meshes. The shape function of polygonal finite element method is constructed by Wachspress generalized barycentric coordinate, and by adding some element-wise bubble functions to the finite element solution, we get a new finite volume element solution that satisfies the local conservation law on a certain dual mesh. The postprocessing algorithm only needs to solve a local linear algebraic system on each primary cell, so that it is easy to implement. More interesting is that, a general construction of the bubble functions is introduced on each polygonal cell, which enables us to prove the existence and uniqueness of the post-processed solution on arbitrary convex polygonal meshes with full anisotropic diffusion tensor. The optimal $H^1$ and $L^2$ error estimates of the post-processed solution are also obtained. Finally, the local conservation property and convergence of the new polygonal finite volume element solution are verified by numerical experiments.