We present Hubble Space Telescope ultraviolet and infrared observations of eight fast radio burst (FRB) host galaxies with subarcsecond localizations, including the hosts of three known repeating FRBs. We quantify their spatial distributions and locations with respect to their host galaxy light distributions, finding that they occur at moderate host-normalized offsets of 1.4r e ([0.6, 2.1]r e ; 68% interval) and on fainter regions of their hosts in terms of IR light but overall trace the radial distribution of IR light in their galaxies. The FRBs in our tested distribution do not clearly trace the distributions of any other transient population with known progenitors and are statistically distinct from the locations of LGRBs, H-poor SLSNe, SGRBs, and Ca-rich transients. We further find that most FRBs are not in regions of elevated local star formation rates and stellar mass surface densities in comparison to the mean global values of their hosts. We also place upper limits on the IR flux at the FRB positions of m IR ⪆ 24.8-27.6 AB mag, constraining both satellite and background galaxies to luminosities well below the host luminosity of FRB 121102. We find that 5/8 FRB hosts exhibit clear spiral arm features in IR light, and that the positions of all well-localized FRBs located in such hosts are consistent with their spiral arms, although not on their brightest regions. Our results do not strongly support the primary progenitor channel of FRBs being connected with either the most massive (stripped-envelope) stars or events that require kicks and long delay times (neutron star mergers).