Here's the petition.
"[I]t is exceedingly rare for this Court to grant rehearing. But when this Court has conducted plenary review and then affirmed by vote of an equally divided court because of a vacancy rather than a disqualification, the Court has not infrequently granted rehearing before a full Bench."
The petition cites more than a dozen historical examples, and also provides compelling reasons why this case warrants such an unusual rehearing--in particular, that a nationwide injunction is in place; that the issue will not arise in any other circuit; and that there's no doubt about the outcome of further proceedings below in light of now-binding Fifth Circuit precedent on all the relevant issues, including standing, notice-and-comment, and the merits.
UPDATE: Further details from Amy Howe here.