  We present an analysis of JWST Ly$\alpha$ spectroscopy of $z\gtrsim6.5$
galaxies, using observations in the public archive covering galaxies in four
independent fields (GOODS-N, GOODS-S, Abell 2744, EGS). We measure Ly$\alpha$
emission line properties for a sample of $210$ $z\simeq6.5-13$ galaxies, with
redshifts confirmed independently of Ly$\alpha$ in all cases. We present $3$
new detections of Ly$\alpha$ emission in JWST spectra, including a large
equivalent width (EW $=143\ \r{A}$) Ly$\alpha$ emitter with strong CIV emission
(EW $=21\ \r{A}$) at $z=7.1$ in GOODS-N. We measure the redshift-dependent
Ly$\alpha$ EW distribution across our sample. We find that strong Ly$\alpha$
emission (EW $>25\ \r{A}$) becomes increasingly rare at earlier epochs,
suggesting that the transmission of Ly$\alpha$ photons decreases by $4\times$
between $z\simeq5$ and $z\simeq9$. We describe potential implications for the
IGM neutral fraction. There is significant field to field variance in the
Ly$\alpha$ emitter fraction. In contrast to the three other fields, the EGS
shows no evidence for reduced transmission of Ly$\alpha$ photons at
$z\simeq7-8$, suggesting a significantly ionized sightline may be present in
the field. We use available NIRCam grism observations from the FRESCO survey to
characterize overdensities on large scales around known Ly$\alpha$ emitters in
the GOODS fields. The strongest overdensities appear linked with extremely
strong Ly$\alpha$ detections (EW $>50\ \r{A}$) in most cases. Future Ly$\alpha$
spectroscopy with JWST has the potential to constrain the size of ionized
regions around early galaxy overdensities, providing a new probe of the
reionization process.
