No, that isn't true. The National Guard reports to the President, not the Vice President. The President approved the use of the National Guard the same hour it was requested by the Capitol Police.
At that time the Capitol Police had oversight of the security of the Capitol grounds. They had made no proclamation that an insurrection was happening and made no authorization for the National Guard to act. Any use of the National Guard troops on the Capitol before the Capitol Police requested and authorized them would have been argued to be an insurrection attempt. What doesn't make sense is you arguing the President Trump using the National Guard without authorization to storm the Capitol wouldn't be labeled as an insurrection by his opponents.
According to the January 6th Special Committee report the request for the National Guard was only made during a call at 2:30 and President Trump had authorized the National Guard to help the Capitol Police by 2:56.
"During a 2:30 p.m. call set up by HSEMA Director Rodriguez, the USCP specified the support they needed from the Guard. Mayor Bowser also made “two urgent requests of the President” that she communicated to Chief of Staff Mark Meadows.
168 One was for the National Guard’s assistance. The other was that the President make a statement asking “people to leave, to leave the building and to get out of the city, to stop.” At 2:56 p.m., Meadows told Mayor Bowser that the President “had approved the request” and was “going to make a statement.” "
It also states that Capitol Police Chief Rund did not get authorization from the Capitol Police Board (run by the Sergeant-at-Arms of the House and Senate, that is by proxy then Speaker of the House Pelosi) until 2:10 p.m. on January 6th.
President Trump responded promptly to the request for National Guard troops by the Capitol Police. The delays were the fault of Nancy Pelosi.