I think it’s going to take some effort but should be possible. You could do it with shape keys alone but they are linear and the deformation between extremes can be less than satisfactory.
I don’t use driven shape keys but this looks like the perfect application.
Essentially the shape you have is just like a pair of eyelids. With a pair of bones originating at the sphere’s origin (the true origin as if the spehere was still complete) and extending through the top and bottom “lips” should give you basic control. You’ll need some weight painting to control the extent to which verts are moved since you obviously want the centre of the lips to move more than the corners of the mouth.
Once you’ve got it roughly working you could make some shape keys to smooth the result and set these keys to be driven by the rotation of the two bones.
As I said, I haven’t used drivers yet (I’ve messed with them) but they should do the trick. Ludwig uses driven shapes for facial animation and Bassam did a video presentation on driven shapes for hands at the recent Siggraph conference. Note that adjusting takes some effort since you’ll want to adjust them with the lips closed and Blender doesn’t really like working that way (yet).
Alternatively, a multiple bone setup could do it. Calvin used an interesting setup for a mouth on his free character.