There is an exporting option
Apply Modifiers to
control whether mesh modifiers are applied to the exported mesh.
Exporting mesh shape key is supported, however exporting each shape key is almost like exporting the mesh again, so don't be surprised it takes a relatively long time.
A lot of modifiers are not compatible with shape keys
(e.g. subsurface modifier), so if you found you have
incorrect shape keys exported, try to disable
and do the exporting again. Besides, it is worthwhile to report the
incompatible modifier to the issue list,
which helps to develop the exporter to have a more precise check of modifiers.