Is there a way to do it? When I chech "export shape keys" and reimport the file after the export there's no animation or keyframes but the shape keys are there.
According to the docs it doesn't look like this is possible (at least as of Blender version 2.79).
At the very bottom of that page:
Important things to remember
- only armature animation on mesh, single skin controller
I read that to mean it isn't possible to export shape key animations to .dae from Blender at this time.