There is no judicial remedy. This is a nonjusticeable political question.
There IS a body with Constitutional authority here, but it isn't a court.
Off the top of my head I don’t know, unless you refer to the 10th Amendment rights “reserved by the states or the people”.
Obviously the courts have some authority over the law and some procedures. And they can serve as an intermediary by demanding various steps such as audits of the ballots, revelation of computer software code, handing over evidence, inspection of ballots etc depending on what case(s) they hear and how far down that road they want to go.
At the very least I’d hope the courts take steps to clean up the mess. Mail-voting is going to become the norm, I suspect, and it needs a thorough vetting. The media won’t do it. The public servants won’t do it. The courts can at the very least lift the veil of secrecy that would force, by sheer embarrassment if nothing else, that all states make changes to the procedures and processes. Voter rolls need to be regularly re-confirmed and updated. Ballots need to have a lot of audit trail data encoded in them. And much more than that.