Candy, there are tons of articles available on this very issue. The general consensus is that employers can mandate employees get the COVID vaccine under the EUA and then under the FDA once the vaccinations are approved.
The CDC says:
The federal government does not mandate (require) vaccination for individuals. For some healthcare workers or essential employees, a state or local government or employer, for example, may require or mandate that workers be vaccinated as a matter of state or other law. Check with your employer to see if they have any rules that apply to you.
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The only outs for those that don't want to take it when their employers require it is if they have a medical condition that falls under the ADA and the vaccine would interfere with that disability or they have a deeply held religious belief that doesn't allow them to take vaccines. In either case, they would have to present documentation and then the employer could still deny the exemption if it is an undue burden on the employer.
But hey, if individual MAGA wants to set aside millions to litigate something like this on a case-by-case basis more power to them. The reality is the employers hold all the cards here.
And yes employers can require you to get a flu shot.
Under most circumstances, it is legal for a business to enforce a flu shot policy. In fact, some healthcare companies have even been known to fire employees who refused to be vaccinated with the flu vaccine. While vaccine injury is rare, there are well documented cases of severe and adverse effects in some people (most notably shoulder related injuries, known as SIRVA); the main reason an employee may object to injecting a live weakened virus into their system. Unfortunately for those employees who have a fear of needles or other aversion to vaccines, an employer can still require a flu vaccine in many cases. Source
As for lawyers having to spend years to go through things, this is already settled law. Employers can currently mandate different sorts of vaccines so there is a need to reinvent the legal wheel with the COVID vaccine. If anything the argument is much stronger, given the seriousness of COVID vs the Flu.