Glenn Reynolds analyzes this issue here. And quite well, except that I question his last move and his conclusion.
The argument that the Vice President is a legislative official isn't inherently absurd. [My note: It is actually correct, so obviously not absurd]. The Constitution gives the Vice President no executive powers: The VP's only duties are to preside over the Senate, and to become President if the serving President dies or leaves office. The Vice President really isn't an Executive official, and isn't part of the President's administration the way that other officials are.
Whatever executive power a VP exercises is exercised because it's delegated by the President, not because the VP has it already. So to the extent the President delegates actual power (as opposed to just taking recommendations for action) the VP is exercising executive authority delegated by the President. [And Glenn is right about this subtle point.]
However -- and here's where the claim that Cheney is really a legislative official creates problems for the White House -- it seems pretty clear that the President isn't allowed to delegate executive power to a legislative official, as that would be a separation of powers violation.
Yes, it is the italicized portion that is questionable. It sounds correct that the President can't delegate executive power to a legislative official. But why not? Well, the President can't delegate executive power to a member of Congress. That is true, but that is probably for two reasons. First, there is a specific clause which says that a member of Congress cannot also serve at the same time as an officer of the United States (which involves the exercise of either executive or judicial power). But that prohibition does not apply to the Vice President, only to members of Congress, which the VP is not. Second, the Constitution vests legislative power in the Congress of the United States, but the VP is not a member of Congress. It sounds strange that he is the President of the Senate, but not a member of Congress, but the Constitution is pretty clear on this.
So, can the President delegate executive power to the Vice President? It seems that he can, unless there is some implicit prohibition on delegating executive power to someone who has a legislative office. And I don't necessarily find such a limit in the Constitution. It seems clear that the President could delegate executive power to a judge without infringing on a constitutional limitation. Indeed, judges have performed all kinds of executive functions throughout history, starting with Chief Justice Jay negotiation the Jay Treaty.
In fact, Glenn's argument is more far reaching than one might at first think. If he is right, then Presidents cannot delegate power to VPs, but they appear to have done this regularly in the last generation. It would make this modern practice unconstitutional. Of course, this is not an argument against Glenn's reading -- lots of modern practices are unconstitutional. But it would be significant.
Of course, Glenn's final point is that it was politically stupid for the Bush Administration to take this position. Well, that may be, but if it is true, that is because there are a lot of ignorant people out there.
Update: Glenn notes this post, but does not focus on the most significant point: my argument that it is probably not unconstitutional for the President to delegate executive power to the VP, even if he is a legislative officer.