So I have been reading a few posts on this subject. I started to make a post then deleted as I had not researched it enough. From what I can see there is no EO on the federal registry regarding private company mandate on vaccines.

EO 14043 - vaccine requirement federal employees.

https://www.federalregister.gov/documents/2021/09/14/2021-19927/requiring-coronavirus-disease-2019-vaccination-for-federal-employees

I cannot find any specific codified EO mandating vaccine for private businesses greater than 100 employees. It does not seem to exist. If true, this is a strategy, if it doesn't exist it cannot be overturned in the courts. If anyone finds this EO in the registry or even OSHA rules on it please post.

blog talking about same topic

https://blog.nomorefakenews.com/2021/10/05/wheres-the-biden-executive-order-mandating-the-vaccine/

This could be used as another arrow in your defense for company mandated jab. Large companies won't care about this, but smaller companies between 100 and 500 might.