The Continuous Galerkin Virtual Element Method (CG-VEM) is a recent innovation in spatial discretization methods that can solve partial differential equations (PDEs) using polygonal (2D) and polyhedral (3D) meshes. Recently, a new formulation of CG-VEM was introduced which can construct VEM spaces on polygons with curvilinear edges. This paper presents the application of the curved VEM to the multigroup neutron diffusion equation and demonstrates its benefits over the conventional straight-sided VEM for a number of benchmark verification test cases with curvilinear domains. These domains were constructed using a topological data-structure developed as part of this paper, based on the doubly-connected edge list, with curves and surfaces both represented using non-uniform rational B-splines (NURBS). This data-structure is used both to specify the geometry of the reactor and to represent the curvilinear polygonal mesh. We also present two separate methods of performing integrations on curvilinear polygons, one for homogeneous functions and one for non-homogeneous functions.