"A" is used if there are others. "The" is used if there are no others.
So in this particular context, since there is generally only one vice president in a given organization (company, government, what have you), you would use "the" when talking about the position in a particular circumstance (e.g. a particular company or government), but you would use "a" when talking about vice presidents in general (i.e. inclusive of all groups that have vice presidents).
Generally speaking, "the" would be used here, since the existence of vice presidents in e.g. other companies and governments is irrelevant to the speaker's complaint.