What action can I take against a pediatrician/medical clinic that issued a flu shot for a child that was not supposed to have one?

Was the child injured by the flu shot. If he or she was, and if there was some reason the clinic/doctor should have known to not administer a flu shot, then you may have a medical malpractice claim.

However, if the child was not injured in some way, and you did not incur any significant medical costs as a result of it, there is no point in a lawsuit: the legal system provides compensation only for actual injuries or losses.

In any event, you may try reporting the doctor or clinic either to his/her/their superiors and/or to the state agency which licenses medical practicioners.