Can gingivitis cause swollen lymph glands in neck and groin?

Gingivitis: Gingivitis normally does not cause neck lymph glands swelling, though it is possible. But not a cause of groin swelling. Best to have this checked including full work up.

If it is actively: infected, yes in the neck. But probably not in the groin. If your inguinal and cervical lymph nodes are enlarged, it usually means systemic infection. But it could also mean Hodgkin's disease or lymphoma in rare cases. In any case, it probably needs treatment. See your doctor asap.

Yes and maybe...: Gingivitis will start a response in your immune system that would potentially involve lymph nodes in your neck. Oral infection does not however directly recruit lymph nodes in the groin area. While it's possible you have a systemic condition that is affecting both areas, it may be multiple problems at the same time. Brush & floss daily; see your dentist & your physician.