"Run" is the better word. An errand is not simply a chore or something that needs to be done. An errand is a mission or journey to accomplish something. An errand requires motion. You "send" someone on an errand rather than "ask" them to "do" an errand. Going to the grocery store to buy some vegetables would be an example of an errand. "Do" is not wrong, and it has the correct meaning. However, "doing" does not suggest motion the way that "running" does. Hence "run" is a much better word to use.

Collocations (words that go together) can be largely regional - to do an errand or to run an errand could both be correct although 'run an errand' is generally more common.

A