Can Trump actually close the DOE?
Technically, yes.
However, “It would take an act of Congress to take it out,” Don Kettl, professor emeritus and former dean of the School of Public Policy at the University of Maryland, told Vox. “It would take an act of Congress to radically restructure it. And so the question is whether or not there’d be appetite on the Hill for abolishing the department.”
More.
That's because an act of Congress created it in the first place.
Trump is not a dictator, and never will be, although he plays one on TV, which is the real problem.
It's all just words.