Not sure, but the way I see it, if Nixon can Create it with the Stroke of a Pen. President Trump can eliminate it the same way.
http://www.presidency.ucsb.edu/ws/index.php?pid=2575&st=environmental+protection+agency&st1
The congress will have to be involved since they did not veto Reorg Plan 3 in 1970. The President can’t just eliminate a cabinet level department on his own.