I shot a gray fox one time that showed up in my grandmother's yard and wouldn't leave. It hung around quite a while, acting strange. She phoned me and it took several minutes for me to drive to her place. The fox was still there when I arrived, so I shot it, then used gloves and buried it well.
I've heard some here say that fox can get "drunk" from eating polk berries, which is a possibility for that fox, but I didn't take any chances.
It's hard to say about your groundhog. I have never heard of that before, but who knows.