Keep in mind that your 6 year old will need a hunter safety certificate if he is shooting the groundhogs
He can do the apprentice license with online exam. Its watching a video or three and answering a small number of questions.
If done honestly (IE the parent isn't just taking it for them), I doubt a 6yr old would likely pass it. But up to the parent to determine if they are mature enough with supervision to go out hunting.
One thing to keep in mind is a person may only get an apprentice license ONCE. So if the OP's/OP's son is interesting in continuing hunting they will need to take the full hunter's safety course the following year and get their license, which I doubt any 7yr old could manage. Not that a 7yr old can't be interested and sharp enough to learn the basic safety rules, but their reading would need to be up to par to read through the questions and answer the multiple choice questions. Now some instructors might be okay with it, but I am not aware of a parent/guardian being able to assist a child in taking the exam (IE reading the questions an answers to them).
IMHO, to get the hunter's safety license you really should be somewhere around 10 or 11 (maybe 9). OP, I'd concentrate on taking your 6yr old WITH you hunting, rather then getting them in to being the hunter due to the roadblocks in this state (well, most states) to getting a young child legally eligible to hunt.
*edit* Just saw your other response. Honestly a better option is probably just take him squirrel hunting this fall. A much higher chance of success.