How might I get hold of a dataset from an old paper?
Lots of old datasets still exist and are stored in repositories, exactly to solve the problem you face. So my first port of call would be to look there. Which one depends on your discipline.

If that is not the case you can try track down her or his PhD students from that time. Chances are they worked with that data, and may know what happened to it. 

Sometimes data is just lost. I remember a case where data was stored on punch cards in a university cellar, and there were also mice that liked eating cardboard in that cellar...