There is no moral, legal, constitutional, or scientific authority for this if one is ever actually issued. As of October 20, 2021, there is no Presidential Executive Order mandating employers with 100 or more employees to receive a vaccination. It is figment of the collective imagination fueled by smart phones and computer screens.
As with the income tax, the U.S. federal govermment is attempting to compel employers to do its dirty work by means of lies and threats of fines and imprisonment. See https://www.williambernardbutler.com/set-my-people-free/.
God and Satan both require man's consent. In the case of the U.S. federal goverment, the pattern is clear. When the federal goverment does not have authority in an area in which it needs to gain authority in order to maintain its control: (1) it lies; (2) then it threatens private employers with fines and imprisoment (latest congressional bill threatens employers with up to $70,000 in OSHA fines for unvaccinated employees) if they do not force the lie upon their employees; and (3) fearful employers then consent and submit to the lie, thus expanding federal jurisdiction and control.
The Good News. World War III is a war taking place in the human mind and God, through Christ, has already won. Because we are stuck in time and space, we have difficulty seeing this already accomplished reality. Unlike the income tax in 1943 when no one could scour the Code to identify the subterfuge, accurate information today travels very quickly.
There is no Executive Order. There are no OSHA rules. Federal rulemaking is a process that takes time and requires public input/comment before enactment. If you are an employer of more than 100 people and you don't know what to do, do nothing until you see something actually requiring you to do something. And then, read it very carefully before you do anything.