There is a lot of information that suggests otherwise, but its really difficult to verify since the laws, acts, and EOs that make it all come together are numerous resulting in tens of thousands of pages of stuff to look through. Some of them may be inaccessible to the general public.

The possible reason it is so difficult to verify the real powers of FEMA may be because it was designed to be the Deep States secret weapon (from 1995). Notably Rex 84 (a result of Reagan's "Presidential Director Number 54" (whatever that is)) suggests that FEMA, once enacted can suspend the constitution and has direct control over the military at that time. Looking right now I can find some stuff that looks like corroboration, but nothing definitive. The PD#54 is confusing because "Presidential Directives" were from Carter, the Reagan equivalent were NSDD. A database of these can be found here and additional resources here.

FEMA certainly manages and pays for the National Guard under an emergency declaration. Who the ultimate authority is less clear, though the above suggest its FEMA. FEMA is supposedly under control of the President, as are the NG, but what happens when they are on foreign soil, like D.C.? What happens when the "President" is not considered legitimate? There is a deep rabbit hole here, that warrants further exploration. I still stand by my statement above, even though the evidence is not conclusive.

This gets tricky. It’s not so simple. The funding for potential action could be given through FEMA. Give orders? Probably not. But provide resources in a timely manner without red tape? Absolutely. Read carefully in the EOs. The STORM act in particular.