I had a dog which was a good groundhog hunter. A Lab cross, believe it or not, but she had the teeth and hunting instincts of a terrier. The groundhog problem disappeared completely when coyotes moved into the area.
Get in touch with someone from the JRTCA, who would be willing to bring their Jack Russells and go to ground for you. They bring their own shovels to get the dogs back, but i think you might have to be there to contain the groundhogs, as some JRT's won't kill them. They just hold them in place.
I have been told, but have not verified, that if you dump "used" kitty litter down their holes that they will choose to live somewhere else.
I can verify it!! I've fenced off two fields now that had groundhogs in them. We tried everything, from having someone shoot them, to stopping up the holes and putting poison smoke bombs in their dens. Nothing worked, they dug back out again. I tried the used kitty litter and it WORKED right away. I put it down all the holes I could find, waited a week and then filled in the holes. Four years later, still no sign of any groundhogs (or woodchucks, as we call them here).
Really, try it -- it is effective, cheap, "organic" and even kind -- apparently they are very tidy animals who are repulsed by the disgusting kitty litter so they go elsewhere.
never heard of kitty litter, unfortunately our barn cats don't use a litter box, i need to find someone who has cats!! An old farmer recently told me bubble gum, they eat it but can't digest it and they die....i'm of to Sam's Club to get some double bubble or bazooka joe!!
I saw that there were ground hogs in the neighborhood when we moved in. I have three JRT's and a Lab that have the run of all the fields in front of our house. They have kept them ground hog free! The field in back, we let the GH's get into...but if we saw them setting up housekeeping, we would toss the dogs back there, too.