That presupposes that elections are required for the states ec votes to be distributed. The states could distribute their ec votes based on dick length, and it would be fully constitutional.

There are ways to get voter ID on a national level, but it isn't through direct executive action, but rather through the justice department suing under the equal protection clause, as states that don't require reasonable ID and/or don't have high numbers of illegals have their votes and representation diluted. This is a favorable path because as you may have noticed, the liberal states voters are harmed by not having voter ID, and the conservative states are harmed by not having illegals.

The next easiest path would be for the DOJ to sue states under the interstate commerce clause. It's a little tougher, but not impossible to argue that a lack of voter ID is unduly affecting the economies of some states due to illegals voting.