No law, no authority, but there will always be sheeple who will take their word for it and willingly comply.
For what it's worth, common vernacular conflates "law" and "regulation" into "law." Companies generally comply with OSHA, with non-compliance resulting in fines that the companies pay -- law or no law (or not a law), companies submit to OSHA.
Plenty complain about the regulations too, not referring to the vax thing, but other regulations. Still, for the most part, companies just do what they are told and pass increase in cost to the customer. It's an example of "can't fight city hall." You can fight, but 99 times out of 100 the court will side with city hall.