That's quite a reach to say "because we have federal contracts, you need a vaccine"
There's no provision that says the mere existence of federal contracts in a company means everyone in the company needs vaccinated. It just says anyone working directly or indirectly on a federal contract. It does say that if you are a non-contractor employee sharing workspace with an eligible contractor, you need vaccinated.
Yeah, it's just cover. Private companies are allowed to make vaccination a condition of continued employment. It's certainly litigatable whether gov't could force every person physically present in the US to get injected. It's obviously to an employer's advantage to have a fully-vaccinated workforce. But for the fear that employees would quit and work elsewhere, more employers would have issued a mandate already.
United led the pack, and discovered that while, yes, some employees refused the vaccine and will be separated, other people like the idea of working among an entirely-vaccinated workforce.
That was something of a revelation to many employers, so we'll likely see more employer mandates. You may lose Suzie, but Bob will want her job.
In the meantime, the WH mandate (which hasn't actually been fully-implemented) was designed to make employees think that it's going to be inevitable at ANY employer, so they should go ahead and do it.
The health care experience in NYC is instructive. When given a hard inflexible deadline, all but a very-tiny few end up getting vaccinated. This is because it isn't just one thing that causes people not to want it. Some are afraid of needles. Some are afraid of doctors/shots/etc. Some are afraid that they'll experience a bad side effect. Some are perfectly-comfortable getting a shot, but see no imperative to do so, and are the kind of folks that won't do anything unless they absolutely have to or are given a deadline; otherwise, they'll just keep putting it off no matter how easy you make it for them. Most people won't admit to any of these things, so what they say isn't what's really going on in their head, consciously or unconsciously.
The thing about the Delta variant is this: it's so-significantly-contagious that non-N95 masks and distancing are actually of significantly-attenuated benefit as compared to the ancestral strain of the virus. So whereas you needed maybe 30 minutes of exposure within 6 feet, unmasked, with the original strain to have a good chance of getting it, now the thing will find you with much less exposure. It's more-contagious, by a significant degree, than colds, flu, polio, SARS, chickenpox and a host of others. It's not as contagious as measles (which can infect you two floors away in a building -- crazy contagious). But it's contagious-enough that, unlike the ancestral strain, it can come off one contagious patient and find a lot of people. If you're unvaccinated, it really is at this point a question of when, not whether. So the calculus about whether to get the vaccine really now comes down to "Am I likely to get really sick or die from the thing when I get it." Unfortunately, while high-risk groups are well-defined, it is still taking out lots of folks in healthier, non-obvious populations, and you really don't have the ability to know how sick you're gonna get. For most, it is always a very-unpleasant experience that they don't want to repeat and one that, once they have it, cause them to wish they had been vaccinated. The actual group of "asymptomatic" patients was very-very small; most were probably false-positive tests rather than actually asymptomatic infected people. As a broad generalization, if you get this thing, and are otherwise-healthy, you're gonna feel sicker than you ever have.
For myself, at least, the choice was obvious. For others, the crazy-low incidence of side effects from this vaccine, and the almost-complete absence of very-serious side effects should enter into their calculus, but that requires them to have a reliable source of accurate, contextualized, information, something that social media isn't doing a good job of providing.