Can a dental infection cause bodily itching?

No: Most likely t have another cause of itching.

Dental infection: Yes, it could if it is bad enough. It could also cause a red rash or line in the area which is usually very serious. Once this happens the bacteria is spreading throughout the body causing septicemia and can even cause death from septic shock. If you are on antibiotics for the infection, your itching and rash could be a side effect of the antibiotics. I would seek care from your dentist ASAP.