The commander of the base wrote, "I do not presume to understand current politics, but what I do know is the concerns of the US administration discussed by Vice-President Vance on Friday are not reflective of Pituffik Space Base." in an email and was thus dismissed. In dismissing her, the Pentagon stated this reason. "Actions [that] undermine the chain of command or to subvert President [Donald] Trump's agenda will not be tolerated at the Department of Defense."
That's not what chain of command means. It simply establishes the hierarchy. Commands can be given by anyone who outranks you, or holds a billet that comes with higher authority. Trump and Vance at the top of that chain and can issue legal orders to literally anyone in the US military.