I don't pretend to know if he actually has this authority, but I suspect he doesn't.

He has the authority to sign the EO, of course. And it's been drafted in such a way that the EO itself is probably within his authority - it directs his subordinates to eliminate the DOE as much as possible consistent with the law.

Whether any specific elimination/transfer of activity is consistent with law is subject to dispute, of course - there have already been lawsuits fighting back against the ending/destaffing/suspension of various programs, like some of the low-income student loan repayment programs. And of course, major changes in government policy are often covered by the Administrative Procedure Act - so sometimes even changes that the government is allowed to make have to be done through those more formal procedures.

But as a general matter, if the President directs the Sec of Ed to eliminate everything in the Department that isn't required to be in the Department by law, and to conduct that elimination in accordance with law, that's probably something he's allowed to do. The Department wouldn't be eliminated, absent Congressional action - but it would be stripped of everything it was doing that wasn't required to be in DOE and not some other agency by law.