I want to add a driver to a vertex' z position. Is this possible? If it is, how?
When I try to add a driver, all I get is the option to edit source and edit translation.
In edit mode, select the vertex you want to animate. Then CtrlH > H (Hook to new object). This will create an empty, whose position you can animate, and the selected vertex will follow it.
You could also add Shape Keys and animate the influence, either by adding keyframes directly or by a driver.
To answer the question specifically: Yes. It is possible.
Go to Outliner, switch Display Mode to Datablocks, then you'll see a list of all datablocks for the .blend file.
Expand the list: Blendfile Data > Objects > your_obj_name > Data > mashdata_name > Vertices > Mesh Vertex > Location, then you can right click on X Y Z, and choose Add Drivers or Add Single Driver from the menu.
However, when there are too many vertices, it will be very hard to target the one you want. I would rather use hooks.