Do people ever get an infection at site of local anesth. from dental visit? or sores?

Never happened to me: In over 40 years I have never seen an infection at injection site. One can develop soreness in the area, but almost always mild and short lived.

Your mouth: is very vascular, difficult to get an infection in that manner, and dentist uses sterile needle and medication

Yes, it is possible: Inserting a foreign body into a human body may bring risk of complications and dental local anesthesia is not an exception. It is very rare complication. However, infection in administrating dental local anesthesia may occur due to preventable or unpreventable circumstances.