It’s not actually the executive branch by itself it’s very nuanced:
1. For health care workers, Centers for Medicare and Medicaid services will be revising payment rules and regulation such that in order to get paid your employees need to be vaccinated. This is a so called “power of the purse” move. It has been used by CMS many times before to force things on the healthcare industry (electronic medical records, JACHO certification for hospitals, etc.). Healthcare industry components doesn’t have to comply, but if it doesn’t then It loses its largest payer source. This is a tried and true tactic that has been used for everything from speed limits to minimum drinking age and I don’t see how it’s not going to be held up in court.
2. For large employers OSHA, an executive branch agency imbued with the powers to regulate workplace safety by Congress, is mandating a vaccine or testing procedure. Most of the lawsuits for this one are focusing on if the process for making this rule are being correctly followed, and are centered on the fact (at least initially) if doing so under an emergency order is valid. Meanwhile, OSHA has already started the standard way to apply a new rule that takes longer but will be harder to challenge. If anything has a chance of being overturned it’s this aspect.
3. Finally for employees and contractors of the federal govt the President, acting as the head or boss of those workers has mandated it. Provided it is not discriminatory towards the disabled (must accept medical exemptions,) race/sex (this applies to all so it’s not) or religious beliefs (need to accept religious exemptions) he is well within his power, which is why those cases have all been dismissed.