We present the VLBA Extragalactic Proper Motion Catalog, a catalog of extragalactic proper motions created using archival VLBI data and our own VLBA astrometry. The catalog contains 713 proper motions, with average uncertainties of ∼24 microas/yr, including 40 new or improved proper motion measurements using relative astrometry with the VLBA. We detect the secular aberration drift - the apparent motion of extragalactic objects caused by the Solar System's acceleration around the Galactic Center - at 6.3 sigma significance with an amplitude of 1.69 +/- 0.27 microas/yr and an apex consistent with the Galactic Center (275.2 +\- 10.0 deg, -29.4 +\- 8.8 deg). Our dipole model detects the aberration drift at a higher significance than some previous studies (e.g., Titov & Lambert 2013), but at a lower amplitude than expected or previously measured. We then use the correlated relative proper motions of extragalactic objects to place upper limits on the average velocity of galaxies caused by the mass distribution of large-scale structure (e.g., Quercellini et al 2009, Darling 2013}). Pairs of small-separation objects that are in gravitationally interacting structures, such as virialized clusters, will show a net decrease in angular separation (> - 15 microas/ye) as they move toward each other, while pairs of large-separation objects that are gravitationally non-interacting and move with the Hubble expansion will show no net change in angular separation. With our catalog, we place a 3 sigma limit on the average rate of convergence of large-scale structure of > -17.4 microas/yr for extragalactic objects within 100 comoving Mpc of each other. We also confirm that large-separation objects (> 800 comoving Mpc) move with the Hubble flow to within ∼ 2.3 microas/yr. Finally, we predict that Gaia end-of-mission proper motions will be able to detect the mass distribution of large-scale structure on length scales 9.8 sigma. This future detection will allow a test of the shape of the theoretical mass power spectrum without a reliance on precise distance measurements.