First, to clear up some misinformation,
The Vice President cannot activate the National Guard.
To repeat, THE VICE PRESIDENT CANNOT ACTIVATE THE NATIONAL GUARD.
Any source stating otherwise is either ignorant or applying artistic license in the reporting. (Frankly, I think it's a signficant mark against the credibility of anyone reporting this, but I'll give them the benefit of the doubt.)
The National Guard can only be activated by the State Governor (not applicable for D.C.), by the President, or by the Secretary of Defense, who reports directly to the President.
Support of operations or missions undertaken by the member’s unit at the request of the President or Secretary of Defense.
Title 32 U.S.C. https://www.law.cornell.edu/uscode/text/32/502
The Vice President has no more authority over the Armed Forces than the Secretary of Education.
Thus, either President Trump or his recently appointed Acting Secretary of Defense Miller activated the National Guard. (Or someone conducted a illegal takeover of the military, but that would be an even bigger headline than the riot itself, and there's little reason to thing that it wasn't Trump or Miller plus media narrative.)
So view the statements ahead with that correction in mind.
The NYTimes said that Trump's VP Mike Pence approved the order:
Defense and administration officials said it was Vice President Mike Pence, not President Trump, who approved the order to deploy the D.C. National Guard.
Though it's unclear what statements -- public or private -- they derived this conclusion from.
The Secretary of Defense Christopher Miller did not mention Trump when speaking about his decision:
Chairman Milley and I just spoke separately with the Vice President and with Speaker Pelosi, Leader McConnell, Senator Schumer and Representative Hoyer about the situation at the U.S. Capitol. We have fully activated the D.C. National Guard to assist federal and local law enforcement as they work to peacefully address the situation.
It's unclear whether Secretary Miller's "we" is himself and the President, or a generous "we" towards the Congressional leaders and the VP (despite them not have any actual decision-making power but perhaps influencing the decision).
The Secretary Miller's Chief of Staff said that the President and the the Sec Def talked "multiple times this week about the request for National Guard personnel in DC" and the
During these conversations the President conveyed to the Acting Secretary that he should take any necessary steps to support civilian law enforcement requests in securing the Capitol and federal buildings
NYT likewise reported:
The president had also expressed interest beforehand in calling in the National Guard to hold off anti-Trump counterprotesters
At 3:36PM EST, White House Press Sectary said Trump gave an order for the National Guard and others to secure the Capitol:
’s direction, the National Guard is on the way along with other federal protective services.
We reiterate President Trump’s call against violence and to remain peaceful.
The Press Secretary's announcement of Trump's order was two hours prior to the National Guard's arrival at 5:40PM according to the timeline from the Dept of Defense.
The following day, Trump said the same:
I immediately deployed the National Guard and federal law enforcement to secure the building and expel the intruders.
Unfortunately, it's not clear what other federal law enforcement agencies were involved (e.g. perhaps the Park Police that were deployed in D.C. in Summer 2020). AFAIK any other agencies would not be under control of Secretary Miller, and so would have to have been ordered by Trump or one of his other Secretaries.
An anonymous CNN source said Trump "initially resisted" deploying the National Guard, and Pence helped get them deployed faster. However, it's unclear whether Trump or Miller make the actual order.
Trump, who has proven over the past year to be eager to deploy the National Guard when violence breaks out, initially resisted doing so on Capitol Hill Wednesday as a mob of his supporters breached the building, per a source familiar. Pence played a key role in coordinating with the Pentagon about deploying them, and urged them to move faster than they were.
There's reason to doubt the White House's claim that President Trump deployed the National Guard. However, and the contrary information is either indirect or secondhand, the inexact reporting about who actually did issue the order doesn't help.
If it wasn't Trump who directly ordered the troop deployment, it was his Acting Secretary of Defense, Christopher Miller.