We present the first X-ray census of fast radio burst (FRB) host galaxies to conduct the deepest search for active galactic nuclei (AGN) and X-ray counterparts to date. Our sample includes seven well-localized FRBs with unambiguous host associations and existing deep Chandra observations, including two events for which we present new observations. We find evidence for AGN in two FRB host galaxies based on the presence of X-ray emission coincident with their centers, including the detection of a luminous (L X ≈ 5 × 1042 erg s−1) X-ray source at the nucleus of FRB 20190608B’s host, for which we infer an SMBH mass of M BH ∼ 108 M ⊙ and an Eddington ratio L bol/L Edd ≈ 0.02, characteristic of geometrically thin disks in Seyfert galaxies. We also report nebular emission-line fluxes for 24 highly secure FRB hosts (including 10 hosts for the first time), and assess their placement on a BPT diagram, finding that FRB hosts trace the underlying galaxy population. We further find that the hosts of repeating FRBs are not confined to the star-forming locus, contrary to previous findings. Finally, we place constraints on associated X-ray counterparts to FRBs in the context of ultraluminous X-ray sources (ULXs), and find that existing X-ray limits for FRBs rule out ULXs brighter than L X ≳ 1040 erg s−1. Leveraging the CHIME/FRB catalog and existing ULX catalogs, we search for spatially coincident ULX-FRB pairs. We identify a total of 28 ULXs spatially coincident with the localization regions for 17 FRBs, but find that the DM-inferred redshifts for the FRBs are inconsistent with the ULX redshifts, disfavoring an association between these specific ULX-FRB pairs.