  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,
$\overline{x}_{HI}$. 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 $\overline{x}_{HI} \lesssim 0.4$ 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: $\overline{x}_{HI}(z=6.6)=0.08^{+ 0.08}_{- 0.05}, \,
\overline{x}_{HI}(z=7.0)=0.28 \pm 0.05$ and $\overline{x}_{HI}(z=7.3)=0.69 \pm
0.11$. 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.
