We present precise measurements of radial (RV) and projected rotational (vsini) velocities of a sample of 258 M6 to L2 dwarfs with multiepoch, high-resolution ({lambda}/{Delta}{lambda}=22500), near-infrared (1.514-1.696{mu}m) spectroscopic observations reported in the Apache Point Observatory Galactic Evolution Experiment Data Release 17. The spectra were modeled using a Markov Chain Monte Carlo forward-modeling method, which achieved median precisions of {sigma}_RV_=0.4km/s and {sigma}_vsini_=1.1km/s. One-half of our sample (138 sources) are previously known members of nearby young clusters and moving groups, and we identified three new kinematic members of the Argus or Carina Near moving groups, 2MASS J05402570+2448090, 2MASSJ14093200+4138080, and 2MASS J21272531+5553150. Excluding these sources, we find that the majority of our sample have kinematics consistent with the Galactic thin disk, and 11 sources are associated with the intermediate thin/thick disk. The field sample has a velocity dispersion of 38.2+/-0.3km/s, equivalent to an age of 3.30+/-0.19Gyr based on empirical age-velocity dispersion relations, and a median vsini of 17km/s. For 172 sources with multiepoch observations, we identified 37 as having significant RV variations, and determined preliminary orbit parameters for 26 sources with four or more epochs, nine of which are short-period binary candidates. For 40 sources with photometric variability periods from the literature less than 5 days and vsini>20km/s, we find a decline in projected radii (Rsini) with age congruent with evolutionary models. Finally, we also present multiepoch RV and vsini measurements for additional 444 candidate ultracool dwarfs.