As the documentation describes, deformation keys store the vertices of the whole mesh, not just parts. There's currently no way to define deformation keys that only affect a subset of the vertices, so there's no way you could just have a face mesh and combine mouth deformation and eye deformation independently using free-form deformation.
Instead, if you really want/need to do this through free-form deformation, you can make these parts into different images and different meshes.
Or, you can keep them as one mesh and deform them using bone Weights.
You can weight the eyes on certain bones, and the mouth on certain bones and the bones can be moved independently.
At the same time, they can be parented to shared bones, so they can move or rotate together as well depending on your bone arrangement.