We present 3D velocity measurements and acceleration limits for stars within ~1 parsec of the Galactic Center black hole, Sgr A*, based on observations of 43 and 86 GHz circumstellar maser emission. Observations were taken with the Very Large Array (VLA) in 2013 and 2014, and with the Atacama Large Millimeter/submillimeter Array (ALMA) in 2015 and 2017. We fit maser positions and spectra to the calibrated visibility data directly in order to mitigate any systematic differences between telescopes and epochs. Combining these data with extant maser astrometry, we calculate stellar proper motions and accelerations with uncertainties as low as ~0.1 mas/yr and 0.01 mas/yr2, respectively, corresponding to approximately 4 km/s and 0.4 km/s/yr at a distance of 8.2 kpc. Doppler tracking additionally enables line-of-sight velocity measurements, though the precision and accuracy of such measurements for deducing the underlying stellar velocities are limited by the complex spectral profiles of some masers. We therefore find acceleration limits typically between ~0.1 to 1 km/s/yr. Upcoming additional epochs from both VLA and ALMA will improve the precision of measured velocities and accelerations. With stellar accelerations, one may infer the 3D position of stars with respect to Sgr A* and therefore measure the stellar and dark matter mass distribution at various radii in the Galactic Center. General relativistic tests may also be possible, such as tests of the equivalence principle or “no hair” theorem. The authors acknowledge support from the NSF Graduate Research Fellowship Program and the NSF grant AST-1908122.