We present a search for RR Lyrae stars using the full six-year data set from the Dark Energy Survey covering ∼5000 deg2 of the southern sky. Using a multistage multivariate classification and light-curve template-fitting scheme, we identify RR Lyrae candidates with a median of 35 observations per candidate. We detect 6971 RR Lyrae candidates out to ∼335 kpc, and we estimate that our sample is >70% complete at ∼150 kpc. We find excellent agreement with other wide-area RR Lyrae catalogs and RR Lyrae studies targeting the Magellanic Clouds and other Milky Way satellite galaxies. We fit the smooth stellar halo density profile using a broken-power-law model with fixed halo flattening (q = 0.7), and we find strong evidence for a break at ${R}_{0}={32.1}_{-0.9}^{+1.1}\,\mathrm{kpc}$ with an inner slope of ${n}_{1}=-{2.54}_{-0.09}^{+0.09}$ and an outer slope of ${n}_{2}=-{5.42}_{-0.14}^{+0.13}$ . We use our catalog to perform a search for Milky Way satellite galaxies with large sizes and low luminosities. Using a set of simulated satellite galaxies, we find that our RR Lyrae-based search is more sensitive than those using resolved stellar populations in the regime of large (rh ≳ 500 pc), low-surface-brightness dwarf galaxies. A blind search for large, diffuse satellites yields three candidate substructures. The first can be confidently associated with the dwarf galaxy Eridanus II. The second has a distance and proper motion similar to the ultrafaint dwarf galaxy Tucana II but is separated by ∼5 deg. The third is close in projection to the globular cluster NGC 1851 but is ∼10 kpc more distant and appears to differ in proper motion.