The stellar velocity distribution function in the solar vicinity is reexamined using data from the Sloan Digital Sky Survey Apache Point Observatory Galactic Evolution Experiment (APOGEE) survey's DR16 and Gaia DR2. By exploiting APOGEE's ability to chemically discriminate with great reliability the thin-disk, thick-disk, and (accreted) halo populations, we can, for the first time, derive the three-dimensional velocity distribution functions (DFs) for these chemically separated populations. We employ this smaller but more data-rich APOGEE+Gaia sample to build a data-driven model of the local stellar population velocity DFs and use these as basis vectors for assessing the relative density proportions of these populations over the 5 < R < 12 kpc and -1.5 < z < 2.5 kpc range as derived from the larger, more complete (i.e., all-sky, magnitude-limited) Gaia database. We find that 81.9% ± 3.1% of the objects in the selected Gaia data set are thin-disk stars, 16.6% ± 3.2% are thick-disk stars, and 1.5% ± 0.1% belong to the Milky Way stellar halo. We also find the local thick-to-thin-disk density normalization to be ρT(R☉)/ρt(R☉) = 2.1% ± 0.2%, a result consistent with, but determined in a completely different way from, typical star-count/density analyses. Using the same methodology, the local halo-to-disk-density normalization is found to be ρH(R☉)/(ρT(R☉) + ρt(R☉)) = 1.2% ± 0.6%, a value that may be inflated due to the chemical overlap of halo and metal-weak thick-disk stars.