The President has no such authority.
But Congress does: Article I section 4:
“The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by Law make or alter such Regulations, except as to the Places of chusing Senators”
From the article it looks like the EO is about denying federal funding for elections in states that don’t have proof of citizenship requirements.
“they will suffer severe cuts in federal funding”
What is amazing is we have Democrats endorsing policies that allow non-citizens to vote. That’s the other side to this story.