If I understand correctly, the White House is now saying that the “dual role” argument is moot, as it is precisely because the VP is part of the Executive, and is part and parcel of the Office of the President, and therefore is part of the enforcement arm of the President’s Executive Orders, that the EO in question doesn’t apply to the VP.
From an article in the NYTimes:
In an interview, a White House spokesman, Tony Fratto, said the executive order explicitly placed Mr. Cheney on equal footing with the president, who was the issuer and enforcer of the order, regardless of any other constitutional questions.
Speaking of the oversight office’s approaches to the vice president’s office, Mr. Fratto said, “It’s not appropriate for a subordinate office like that to investigate or require reporting from the enforcer of the executive order.”
A White House official placed further distance from the dual role argument by adding that Mr. Cheney did not necessarily agree with it.
You can’t make this stuff up.