Someone can run the place under the DOJ rules of succession that have been in place for years. That would be Deputy U.S. Attorney General Rod Rosenstein. That's the whole point of having those succession rules in the first place.
President Trump chose to override the DOJ succession rules. That's his right as the President of the United States. That doesn't mean he can appoint ANYONE to a role that requires Senate confirmation, though.
There is no reason that DOJ rules of succession would take precedence over Congressional legislation creating actual, legislated, rules of succession.
“President Trump chose to override the DOJ succession rules. That’s his right as the President of the United States.”
Well, at least you got that right.
“That doesn’t mean he can appoint ANYONE to a role that requires Senate confirmation, though.”
Were did you get THAT from ?