So.. this says a lot to me. The links might not work since this page likes to put dashes in things but we shall try.
Biden gave an EO on Sep 9th. The EO states that there will be a task force established ("Safer Federal Workforce") that will work on establishing "guidance". It then directs federal agencies to put contractual language requiring the contractor to comply with the Safer Federal Workforce Guidance. The EO can be found: https://www.whitehouse.gov/briefing-room/presidential-actions/2021/09/09/executive-order-on-ensuring-adequate-covid-safety-protocols-for-federal-contractors/
Safer Federal Workforce Guidance put out guidance basically saying all contractors needed to be fully vaccinated by Dec 8th: https://www.saferfederalworkforce.gov/downloads/Draft%20contractor%20guidance%20doc_20210922.pdf
So they're trying to roll it out via contractual terms in federal contracts (starting Dec 8th) RATHER than putting it into law (which would immediately be challenged in the courts). This way, it won't ever see any court room until its put into a contract Dec 8th (and federal contractors can go "but we're just doing what is being told of us"). Once Dec 8th hits, you can expect companies to sue..