They are saying that the mandate for large businesses to covid test non vaccinated employees is not enforceable by the federal DOL? Then they are either uniformed about the law or simply pandering to their ultra conservative base. The Supreme Court in 1905 upheld a Massachusetts vaccination mandate, stating:
It is within the police power of a State to enact a compulsory vaccination law, and it is for the legislature, and not for the courts, to determine in the first instance whether vaccination is or is not the best mode for the prevention of smallpox and the protection of the public health.
Jacobson v. Massachusetts, 197 U.S. 11, 12 (1905). Justice Harlan, who delivered the Court's opinion, went on in the decision to explain:
But the liberty secured by the Constitution of the United States to every person within its jurisdiction does not import an absolute right in each person to be, at all times and in all circumstances, wholly freed from restraint. There are manifold restraints to which every person is necessarily subject for the common good. On any other basis, organized society could not exist with safety to its members. Society based on the rule that each one is a law unto himself would soon be confronted with disorder and anarchy. Real liberty for all could not exist under the operation of a principle which recognizes the right of each individual person to use his own, whether in respect of his person or his property, regardless of the injury that may be done to others.
Id. at 26. Thus, there is no Constitutional limitation on the government here. The only issue then is whether federal statute empowers the federal government to do it. In that regard, Section 6 of the Occupational Safety and Health Act gives broad authority to the Occupational Safety and Health Administration (OSHA) to regulate employers for the purpose of ensuring worker safety. As a result, I think it very likely that the courts will uphold this particular initiative as being within the power of the federal government, the uniformed opinions of some politicians notwithstanding.