Can I splice grounding wires together to reach an old outlet that needs a ground?
As long as the wire is sound and is joined old and new properly, i.e., not just twisted together but secured with a twist lock or crimp connector it meets code and is fine.  Consider that the hot and neutral is already joined there so the ground connection does not degrade that circuit and is quite proper. 
That said - anytime you can replace old wiring with new is a good thing.  If you are running a wire from the panel - I would recommend you run entirely new nm or romex cable as far as you can.