This isn’t part of the mandate. This is related to their federal contracts.
All federal contractors can be told to vaccinate.
That is a material modification of their contracts. The government can’t do that unless the courts back them up, which is possible. But it now opens up contracts to renegotiation in many areas. And it sets a precedent that the government can alter a contract arbitrarily after it is awarded. Should make for some interesting effects on contract law.