Adding holes to this sphere I would like to subtract cylindrical holes in this object, centered over each of the highlighted vertices and aligned with the object's center.  Can somebody show me how to do this?
Steps:

First of all activate LoopTools add-on in User Preferences, than:


CTRL+Shift+B to Bevel vertices.
MouseWheel to add more cuts or press F6 to see menu with Bevel options.
W > LoopTools > Circle.
CTRL+Num -  to select less.
X > Delete Vertices.




Edit:

For OP model holes will have different number of vertices. But besides that everything is working just fine.



As a workaround you can first select vertices with 3 connected edges and make for example 5 Bevel cuts: 3 edges x 5 cuts = 15 vertices.

Than do the same thing with vertices with 5 edges connected with 3 cuts: 5 edges x 3 cuts = 15 vertices.