  We calculate the abundance of UV-bright galaxies in the presence of
ultralight axion (ULA) dark matter (DM), finding that axions suppress their
formation with a non-trivial dependence on redshift and luminosity. We set
limits on axion DM using both Planck cosmic microwave background (CMB) and UV
luminosity function (UVLF) data. We exclude a single axion as all the DM for
$m_{ax} < 10^{-21.6}$ eV and limit axions with $-26 < \log( m_{ax}/{eV}) < -23$
to be less than $22\%$ of the DM (both limits at $95\%$ credibility). These
limits use UVLF measurements from 24,000 sources from the Hubble Space
Telescope (HST) that probe small-scale structure at redshifts $4 < z < 10$. We
marginalize over a parametric model that connects halo mass and UV luminosity
that has been shown to match hydrodynamical simulations. Our results bridge a
window in axion mass and DM fraction previously unconstrained by cosmological
data, between large-scale CMB and galaxy clustering and the small-scale
Lyman-$\alpha$ forest. These high-$z$ measurements provide a powerful
consistency check of low-$z$ tests of axion DM, which include the recent hint
for a sub-dominant ULA DM fraction in Lyman-$\alpha$ forest data. We also
consider a sample of 25 spectroscopically-confirmed high-$z$ galaxies from the
James Webb Space Telescope (JWST). We find that these data are consistent with
the HST UVLF assuming $\Lambda$CDM and our flexible parametric model of UV
luminosity. Combining HST and JWST UVLF data does not improve our constraints
beyond HST alone, but future JWST measurements have the potential to improve
these results significantly. We also find an excess of low-mass halos ($< 10^9
M_\odot$) at $z < 3$, which could be probed by sub-galactic structure probes
(e.g., stellar streams, satellite galaxies and strong lensing).
