We present new central stellar velocity dispersions for 484 Sy 1.9 and Sy 2 from the second data release of the Swift/BAT AGN Spectroscopic Survey (BASS DR2). This constitutes the largest study of velocity dispersion measurements in X-ray-selected obscured active galactic nuclei (AGN) with 956 independent measurements of the CaII H and K {lambda}3969, 3934 and MgI {lambda}5175 region (3880-5550{AA}) and the calcium triplet region (8350-8730{AA}) from 642 spectra mainly from VLT/X-Shooter or Palomar/DoubleSpec. Our sample spans velocity dispersions of 40-360km/s, corresponding to 4-5 orders of magnitude in black hole mass (M_BH_=10^5.5-9.6^M_{sun}_), bolometric luminosity (L_bol_~10^42-46^erg/s), and Eddington ratio (L/L_Edd_~10^-5^ to 2). For 281 AGN, our data and analysis provide the first published central velocity dispersions, including six AGN with low-mass black holes (M_BH_=10^5.5-6.5^M_{sun}_), discovered thanks to high spectral resolution observations ({sigma}_inst_~25km/s). The survey represents a significant advance with a nearly complete census of velocity dispersions of hard X-ray-selected obscured AGN with measurements for 99% of nearby AGN (z<0.1) outside the Galactic plane (|b|>10{deg}). The BASS AGN have much higher velocity dispersions than the more numerous optically selected narrow-line AGN (i.e., ~150 versus ~100km/s) but are not biased toward the highest velocity dispersions of massive ellipticals (i.e., >250km/s). Despite sufficient spectral resolution to resolve the velocity dispersions associated with the bulges of small black holes (~10^4-5^M_{sun}_), we do not find a significant population of super-Eddington AGN. Using estimates of the black hole sphere of influence from velocity dispersion, direct stellar and gas black hole mass measurements could be obtained with existing facilities for more than ~100 BASS AGN.