The Continuous Galerkin Virtual Element Method (CG-VEM) is a recent innovation in spatial discretisation methods that can solve partial differential equations (PDEs) using polygonal (2D) and polyhedral (3D) meshes. This paper presents the first application of VEM to the field of nuclear reactor physics, specifically to the steady-state, multigroup, neutron diffusion equation (NDE). In this paper the theoretical convergence rates of the CG VEM are verified using the Method of Manufactured Solutions (MMS) for a reaction-diffusion problem in the presence of both highly distorted and non-convex elements and also in the presence of discontinuous material data. Finally, numerical results for the 2D IAEA and the 2D C5G7 industrial nuclear reactor physics benchmarks are presented using both block-Cartesian and general polygonal meshes.