  The time frame in which hydrogen reionization occurred is highly uncertain,
but can be constrained by observations of Lyman-alpha (Ly$\alpha$) emission
from distant sources. Neutral hydrogen in the intergalactic medium (IGM)
attenuates Ly$\alpha$~photons emitted by galaxies. As reionization progressed
the IGM opacity decreased, increasing Ly$\alpha$~visibility. The galaxy
Ly$\alpha$~luminosity function (LF) is thus a useful tool to constrain the
timeline of reionization. In this work, we model the Ly$\alpha$~LF as a
function of redshift, $z=5-10$, and average IGM neutral hydrogen fraction,
$\overline{x}_\textsc{hi}$. We combine the Ly$\alpha$~luminosity probability
distribution obtained from inhomogeneous reionization simulations with a model
for the UV LF to model the Ly$\alpha$~LF. As the neutral fraction increases,
the average number density of Ly$\alpha$~emitting galaxies decreases, and are
less luminous, though for $\overline{x}_\textsc{hi} \lesssim 0.4$ there is only
a small decrease of the Ly$\alpha$~LF. We use our model to infer the IGM
neutral fraction at $z=6.6, 7.0, 7.3$ from observed Ly$\alpha$~LFs. We conclude
that there is a significant increase in the neutral fraction with increasing
redshift: $\overline{x}_\textsc{hi}(z=6.6)=0.08^{+ 0.08}_{- 0.05}, \,
\overline{x}_\textsc{hi}(z=7.0)=0.28 \pm 0.05$ and
$\overline{x}_\textsc{hi}(z=7.3)=0.83^{+ 0.06}_{- 0.07}$. We predict trends in
the Ly$\alpha$~luminosity density and Schechter parameters as a function of
redshift and the neutral fraction. We find that the Ly$\alpha$~luminosity
density decreases as the universe becomes more neutral. Furthermore, as the
neutral fraction increases, the faint-end slope of the Ly$\alpha$~LF steepens,
and the characteristic Ly$\alpha$~luminosity shifts to lower values, concluding
that the evolving shape of the Ly$\alpha$~LF -- not just its integral -- is an
important tool to study reionization.
