system:“10:31
Vice president has cleared to us to intercept tracks of interest and shoot them
down if they do not respond per [General Arnold].”

9/11 commission report page 42

The attentive reader will note that I am reading the chapters out of order. 

I strongly suspect that Cheney gave that order on his own (non) authority, although he,  Bush and Rice deny this. 

Cheney claims
that he called the president around 10:00 AM to discuss the issue and the president authorised fighters to shoot down hijacked planes that would not divert.  Oddly, unlike other calls, this call was not logged.  Nor does it appear in the notes taken by Lynn Cheney or Scooter Libby.  Most oddly of all, Joshua Bolton asked Cheney to call Bush to confirm the order after Cheney had authorised shooting down airliners. 

I think the following quote from pages 39 and 40 of the report make it obvious (in case we didn't know already) who was running things then.

The Vice Preesident remembered placing a call to the President just after
entering the shelter conference room. There is conflicting evidence about
when the Vice President arrived in the shelter conference room.We have concluded,
from the available evidence, that the Vice President arrived in the room
shortly before 10:00,perhaps at 9:58.The Vice President recalled being told, just
after his arrival, that the Air Force was trying to establish a combat air patrol
over Washington.213
The Vice President stated that he called the President to discuss the rules of
engagement for the CAP. He recalled feeling that it did no good to establish
the CAP unless the pilots had instructions on whether they were authorized
to shoot if the plane would not divert. He said the President signed off on that
concept. The President said he remembered such a conversation, and that it
reminded him of when he had been an interceptor pilot.The President emphasized
to us that he had authorized the shootdown of hijacked aircraft.214
The Vice President’s military aide told us he believed the Vice President
spoke to the President just after entering the conference room, but he did not
hear what they said. Rice, who entered the room shortly after the Vice President
and sat next to him, remembered hearing him inform the President,“Sir,
the CAPs are up. Sir, they’re going to want to know what to do.” Then she
recalled hearing him say,“Yes sir.” She believed this conversation occurred a
few minutes, perhaps five, after they entered the conference room.215
We believe this call would have taken place sometime before 10:10 to 10:15.
40 THE 9/11 COMMISSION REPORT
Among the sources that reflect other important events of that morning, there is no documentary evidence for this call, but the relevant sources are incomplete.  Others nearby who were taking notes, such as the Vice President’s chief of staff, Scooter Libby, who sat next to him, and Mrs. Cheney, did not note a call between the President and Vice President immediately after the Vice President entered the conference room.216
At 10:02, the communicators in the shelter began receiving reports from
the Secret Service of an inbound aircraft—presumably hijacked—heading
toward Washington.That aircraft was United 93.The Secret Service was getting
this information directly from the FAA.The FAA may have been tracking
the progress of United 93 on a display that showed its projected path to
Washington, not its actual radar return.Thus, the Secret Service was relying on
projections and was not aware the plane was already down in Pennsylvania.217
At some time between 10:10 and 10:15, a military aide told the Vice President
and others that the aircraft was 80 miles out. Vice President Cheney was
asked for authority to engage the aircraft.218 His reaction was described by
Scooter Libby as quick and decisive, “in about the time it takes a batter to
decide to swing.” The Vice President authorized fighter aircraft to engage the
inbound plane. He told us he based this authorization on his earlier conversation
with the President.The military aide returned a few minutes later, probably
between 10:12 and 10:18, and said the aircraft was 60 miles out. He again
asked for authorization to engage.The Vice President again said yes.219
At the conference room table was White House Deputy Chief of Staff
Joshua Bolten. Bolten watched the exchanges and, after what he called “a quiet
moment,”suggested that the Vice President get in touch with the President and
confirm the engage order. Bolten told us he wanted to make sure the President
was told that the Vice President had executed the order. He said he had
not heard any prior discussion on the subject with the President.220
The Vice President was logged calling the President at 10:18 for a two minute
conversation that obtained the confirmation. On Air Force One, the
President’s press secretary was taking notes; Ari Fleischer recorded that at
10:20, the President told him that he had authorized a shootdown of aircraft
if necessary.