It depends upon where you look. You can find plenty of the typical leftist fact checking sites saying it isn't true and saying she doesn't have that authority. Authority doesn't seem to matter that much in the past half decade. There is authoritarian overreach happening all over the place.

The actual report was that both Nancy Pelosi and Mitch McConnell refused National Guard Support.

The Senate security officials do answer directly to those two.

It was reported by Washington DC Police Chief Sund.

It was also reported by Former Sergeant at Arms Paul Irving.

It is a matter of who you trust I guess because the fact checking sites these days are worthless but me giving you other information can seem just as worthless as these days it is easy to find data supporting whatever narrative you are looking for.

I also recall Trump talking about it before January 6th. My memory could be wrong though.