Precise measurements of radial (RV) and rotational (vsini) velocities of stars are essential for studying stellar kinematics (space velocities and dispersions), binary orbits, and rotational dynamics (angular momentum evolution). The high-resolution spectroscopic observations necessary to make these measurements are challenging for the intrinsically faint and low temperature ultracool dwarfs, stellar and sub-stellar objects with masses below 0.1 Msun. Current velocity samples of these objects are small and incomplete. We present a radial and rotational velocity survey of 33 nearby (d ≤ 20 pc) T dwarfs based on forward modeling analysis of nearly twenty years of high-resolution spectra obtained with Keck/NIRSPEC. We present the distributions of our RV and vsini measurements, and report the identification of two new substellar binaries. We statistically analyze kinematic ages of the local ultracool dwarf population based on RV and astrometric measurements of 166 late-M, L, and T dwarfs within 20 pc, and show that the L dwarfs as a population are significantly more dispersed and hence older than expected from population simulations.