Judging by Klain's statement, looks like he's the one who gave this clunker the go-ahead.
Nothing in the word 'occupational' indicates it can be legally tailored to mean over 100 employees. Thus OSHA has no power to enforce anything related to a vaccine-mandate, under the Constitution.
Like anything involving the federal government, things get squirrelly. The magic number is generally 10 - that being the minimum number under which OSHA can enforce workplace safety regulations.
I've been involved with OSHA and safety programs where I work (two different companies) since 2007, so I have a bit of background here.
Where OSHA overextended itself pertains to the General Duty Clause of the OSHA Act of 1970. That General Duty Clause, much like the Constitution's Commerce Clause, has been bastardized and overused to warrant expansion of federal power. In this case, OSHA has gone beyond its own congressional mandate, the courts have recognized it, so DOL has had to back down -- a momentous happening, I might add.
https://yourbusiness.azcentral.com/companies-required-meet-osha-regulations-2788.html