Currently we don’t support blendshapes in our skinned mesh particle support. (We will look at adding it in the future)
As a workaround, it should be possible to use BakeMesh in script, and then assign the result of that as the source mesh. Remember to set the shape type to Mesh, (not SkinnedMeshRenderer), and set the Mesh property in the Shape module, using the BakeMesh result.