My previos response was based on what the OP had posted up to that point. Granted, the OP did say in the original post:
But that seemed to be a desription of the organization's custom, rather than a bylaws provison. If that is what the bylaws actually say, as seems to be the case based on the OP's later response, then I would agree that even though it is not very well written, it probably does mean that the VP doesn't move up, and there is an election for president. I would feel more confident, however, if I could be sure that the langauge is an actual quote, or at least an accurate paraphrase, and not just someone's interpretation of the bylaws.