Could everyone have a gag reflex?

Come again?: With very few exceptions, everyone DOES have a gag reflex. The afferent limb of the gag reflex is mediated by the glossopharyngeal nerve and the efferent limb by the vagus nerve. The reflex is part of your defense against foreign bodies getting lodged in your throat or airway. Maybe you had a different question in mind, because the one you posted doesn't make much sense.