We propose a scheme for achieving widefield coherent anti-Stokes Raman scattering (CARS) microscopy images with sub-diffraction-limited resolution. This approach adds structured illumination to the widefield CARS configuration [Applied Physics Letters 84, 816 (2004)]. By capturing a number of images at different phases of the standing wave pattern, an image with up to three times the resolution of the original can be constructed. We develop a theoretical treatment of this system and perform numerical simulations for a typical CARS system, which indicate that resolutions around 120 nm are obtainable with the present scheme. As an imaging system, this method combines the advantages of sub-diffraction-limited resolution, endogenous contrast generation, and a wide field of view.