Article I Section 4 Clause 1:
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.
Seems pretty clear to me. I see nothing about courts or non-legislative elected or appointed officials setting election rules. Had they been granted those powers, they would have been mentioned.
