So far the executive order and mandate are threats only, and you can’t sue a threat. From Sundance at theconservativetreehouse.com:
Joe Biden did it again today. A month after the first announcement, the White House occupant claimed again a Dept of Labor rule (via OSHA) is forthcoming, yet no such process appears to be taking place. This ploy now seems very purposeful, because without an actual policy or regulation visibly in place, state attorneys general cannot file a lawsuit or request an injunction.
As long as Biden keeps threatening a DOL worker vaccination rule sometime in the future, many employers will take action to require worker vaccination. This seems to be the actual strategy; bolstered by White House Press Secretary Jen Psaki caught off-guard last week when asked about it.
Psaki had no idea how to answer the question about any OSHA activity not taking place (link). Obviously Psaki didn’t expect the question, but it was also obvious that no background conversation had ever taken place amid the White House communication team.