This little video backgrounder comes from the Missouri Dept. of the Conservation.
On another page, the same Department notes that groundhogs "feed in hay or crop fields, home gardens, orchards and nurseries; and the burrows and mounds may be hazardous to farm equipment or to horses and riders. For these reasons, control measures may be necessary."
Instructions are given on how to shoot, use gas cartridges, and trap. Of course, nothing is as targetted or as humane as terrier work.