We present the Virgo Environment Traced in CO (VERTICO) survey, a new effort to map ^12^CO(2-1), ^13^CO(2-1), and C^18^O(2-1) in 51 Virgo Cluster galaxies with the Atacama Compact Array, part of the Atacama Large Millimeter/submillimeter Array. The primary motivation of VERTICO is to understand the physical mechanisms that perturb molecular gas disks, and therefore star formation and galaxy evolution, in dense environments. This first paper contains an overview of VERTICO's design and sample selection, ^12^CO(2-1) observations, and data reduction procedures. We characterize global ^12^CO(2-1) fluxes and molecular gas masses for the 49 detected VERTICO galaxies, provide upper limits for the two nondetections, and produce resolved ^12^CO(2-1) data products (median resolution=8"~640pc). Azimuthally averaged ^12^CO(2-1) radial intensity profiles are presented along with derived molecular gas radii. We demonstrate the scientific power of VERTICO by comparing the molecular gas size-mass scaling relation for our galaxies with a control sample of field galaxies, highlighting the strong effect that radius definition has on this correlation. We discuss the drivers of the form and scatter in the size-mass relation and highlight areas for future work. VERTICO is an ideal resource for studying the fate of molecular gas in cluster galaxies and the physics of environment-driven processes that perturb the star formation cycle. Upon public release, the survey will provide a homogeneous legacy data set for studying galaxy evolution in our closest cluster.