We present the H i mass inventory for the REsolved Spectroscopy Of a Local VolumE (RESOLVE) survey, a volume-limited, multi-wavelength census of >1500 z = 0 galaxies spanning diverse environments and complete in baryonic mass down to dwarfs of ∼109 M⊙. This first 21 cm data release provides robust detections or strong upper limits (1.4M H i < 5%-10% of stellar mass M ∗) for ∼94% of RESOLVE. We examine global atomic gas-to-stellar mass ratios (G/S) in relation to galaxy environment using several metrics: group dark matter halo mass M h, central/satellite designation, relative mass density of the cosmic web, and distance to the nearest massive group. We find that at fixed M ∗, satellites have decreasing G/S with increasing M h starting clearly at M h ∼ 1012 M⊙, suggesting the presence of starvation and/or stripping mechanisms associated with halo gas heating in intermediate-mass groups. The analogous relationship for centrals is uncertain because halo abundance matching builds in relationships between central G/S, stellar mass, and halo mass, which depend on the integrated group property used as a proxy for halo mass (stellar or baryonic mass). On larger scales G/S trends are less sensitive to the abundance matching method. At fixed M h ≤ 1012 M⊙, the fraction of gas-poor centrals increases with large-scale structure density. In overdense regions, we identify a rare population of gas-poor centrals in low-mass (M h < 1011.4 M⊙) halos primarily located within ∼1.5× the virial radius of more massive (M h > 1012 M⊙) halos, suggesting that gas stripping and/or starvation may be induced by interactions with larger halos or the surrounding cosmic web. We find that the detailed relationship between G/S and environment varies when we examine different subvolumes of RESOLVE independently, which we suggest may be a signature of assembly bias.