The dark matter subhalo mass function is a promising way of distinguishing between dark matter models. While cold dark matter predicts halos down to Earth-sized masses, other dark matter models typically predict a cutoff in the subhalo mass function. Thus a definitive detection or limits on the existence of subhalos at small masses can give us insight into the nature of dark matter. If these subhalos exist in the Milky Way, they would produce weak lensing signatures, such as modified apparent positions, on background stars. These signatures would generate correlations in the apparent velocities and accelerations of these stars, which could be observable given sufficient astrometric resolution and cadence. The Nancy Grace Roman Space Telescope's exoplanet microlensing survey will be perfectly suited to measuring the acceleration signatures of these halos. Here we forward model these acceleration signatures and explore the Roman Space Telescope's future constraints on lens profiles and populations. We find that the Roman Space Telescope could place competitive bounds on point-source, Gaussian, and Navarro-Frenk-White (NFW) profile lenses that are complementary to other proposed methods. In particular, it could place 95% upper limits on the NFW concentration c200<102.5. We discuss possible systematic effects that could hinder these efforts but argue they should not prevent the Roman Space Telescope from placing strong limits. We also discuss further analysis methods for improving these constraints.