Volatile molecules are critical to terrestrial planetary habitability, yet they are difficult to observe directly where planets form at the midplanes of protoplanetary disks. It is unclear whether the inner ~1AU of disks are volatile-poor or if this region is resupplied with ice-rich dust from colder disk regions. Dust traps at radial pressure maxima bounding disk gaps can cut off the inner disk from these types of volatile reservoirs. However, the trap retention efficiency and atomic composition of trapped dust have not been measured. We present a new technique to measure the absolute atomic abundances in the gas accreting onto T Tauri stars and infer the bulk atomic composition and distribution of midplane solids that have been retained in the disk around the young star TW Hya. We identify near-infrared atomic line emission from gas-phase material inside the dust sublimation rim of TW Hya. Gaussian decomposition of the strongest H Paschen lines isolates the inner disk hydrogen emission. We measure several key elemental abundances, relative to hydrogen, using a chemical photoionization model and infer dust retention in the disk. With a 1D transport model, we determine approximate radial locations and retention efficiencies of dust traps for different elements. Volatile and refractory elements are depleted from TW Hya's hot gas by factors of ~10^2^ and up to 10^5^, respectively. The abundances of the trapped solids are consistent with a combination of primitive Solar System bodies. Dust traps beyond the CO and N_2_ snow line cumulatively sequester 96% of the total dust flux, while the trap at 2AU, near the H_2_O snow line, retains 3%. The high depletions of Si, Mg, and Ca are explained by a third trap at 0.3AU with >95% dust retention. TW Hya sports a significant volatile reservoir rich in C- and N-ices in its outer submillimeter ring structure. However, unless the inner disk was enhanced in C by earlier radial transport, typical C destruction mechanisms and the lack of a C resupply should leave the terrestrial planet-forming region of TW Hya "dry" and carbon-poor. Any planets that form within the silicate dust trap at 0.3 AU could resemble Earth in terms of the degree of their volatile depletion.