No, Trump did not order the National Guard to mobilize at that time. As The New York Times reported, Army Secretary Ryan McCarthy and Acting Defense Secretary Christopher Miller made the ultimate decision to activate the entirety of the district's National Guard, an order that was subsequently approved by Mike Pence, not Trump:
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. It was unclear why the president, who incited his supporters to storm the Capitol and who is still the commander in chief, did not give the order.
The claim that Miller gave the order is borne out by the official timeline of events released by the Department of Defense, which depict a number of conversations between officials between December 31 and the 8:00 PM on January 6. Following a call between McCarthy and the mayor of Washington, D.C., Muriel Bowser, there were additional meetings in which Miller, the Deputy Mayor, and other high-level officials were present. Neither the Vice President or the President are listed as having been in attendance, but it explicitly states that at 3:00 PM, McCarthy determined that the entirety of the district's National Guard forces would be required, and that Miller gave verbal approval for the full deployment at 3:04 PM. Pence's involvement is not, then, fully supported, but it makes any involvement by Trump quite dubious.
(To clear up some confusion: I intentionally went along with article's choice of the verb "approved" and not "ordered" to describe Pence's involvement. As others have correctly mentioned, the Vice President has no authority over the National Guard; here, by "approved", I'm referring to a non-binding order. Pence giving the deployment the okay would effectively just show that the administration is on board with the decision made by Miller and McCarthy - it wouldn't look good if the Department of Defense deployed the National Guard and the White House disapproved of the order.)
CNN, too, has reported that sources have said that Pence, not Trump, was involved. Public statements supporting this were made by Miller and the House Minority Leader, Kevin McCarthy:
Public statements by acting Defense Secretary Christopher Miller and other top officials suggested it was Pence who ultimately approved the decision. Miller's statement Wednesday seems to indicate he did not even speak with Trump, discussing the matter with his deputy instead as sources told CNN the President was reluctant to even denounce the violence being carried out in his name.
. . .
Republican House Minority Leader Kevin McCarthy sidestepped questions Wednesday night about whether Pence, not Trump, directed the DC National Guard to be activated but suggested the vice president ultimately approved the decision.
Asked by Fox News about reporting that Pence, not the President, approved the activation, McCarthy demurred, but ultimately said: "I know the vice president has been in constant contact with us and also along with security inside the Capitol, I communicated with the vice president early on. It was in regards to getting the National Guard there. He said he will call right now."
I should note that, like the DoD's timeline, Miller's statement explicitly mentioned the Vice President and other key officials, but did not refer to the President at all.
However, Miller has since submitted testimony (on May 12) to the House Committee on Oversight and Reform in which he states that he doesn't recall the President being on any of the calls on January 6 in which plans (and presumably decisions) for the National Guard's response were made. Moreover, he says that the President had no part in the decisions made that day by the DoD at all:
I also want to address questions that have been raised in regard to the President’s involvement in the response. He had none with respect to the Department of Defense efforts on January 6.
. . .
My recollection is that White House personnel, including White House counsel and the Chief of Staff, but not the President, participated in some of the multi-agency calls on January 6 convened to organize and coordinate the response to the events at the Capitol after the mob entered the Capitol building. I do not recall any specific remarks they made, if any, during those calls. Those calls are reflected in the various timelines available to you.
This appears to be referencing that DoD timeline, and it does explicitly state that he does not recall Trump ever being involved in the decisions of that afternoon.
The Wednesday decision by Ryan McCarthy and Miller to mobilize the entirety of the district's National Guard contingent was made after an earlier deployment of 350 members of the National Guard on Monday following repeated requests by Bowser - requests that were then approved by the Pentagon. Politico notes that
Miller spoke with Trump multiple times this week about the mayor’s Guard request, said DoD Chief of Staff Kash Patel. During those conversations, Trump conveyed that Miller should take "any necessary steps to support civilian law enforcement requests in securing the Capitol and federal buildings."
These, too, are supported by the DoD's timeline. It's not clear whether Trump ever explicitly mentioned the National Guard, but any rate, the conversations between Miller and Trump seem to have occurred prior to the events of January 6 and the storming of the Capitol.
Yes, the White House has claimed that Trump gave the order, as per a tweet by White House Press Secretary Kayleigh McEnany:
At President @realDonaldTrump’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.
In addition, in a video released on January 7, Trump himself claimed
I immediately deployed the National Guard
However, Trump's claim is dubious at best:
- It contradicts the unnamed sources cited by The New York Times and CNN, as well as public statements by Miller and (the more oblique one) Minority Leader McCarthy.
- The Department of Defense's timeline, which explicitly mentions conversations with Trump involving the National Guard earlier in the week, does not mention him in connection with any relevant meetings on Wednesday - rather odd, if Trump had been the one to give the order.
The White House is the only government source I'm aware of claiming that Trump gave the command, and many others involved have said otherwise.