They should have the same liability as to anyone which interacts with the company or organizations, including patients outcomes in hospitals. If a patients get infected by a sick doctor or nurse, the fault is at the employer who allowed that situation from happening.

Every person who get injured at work is to some degree harmed by the company that allowed that injury to occur. This is why companies has insurance to deal with that. If vaccinations is required for the job then getting vaccinations is by definition work, should happen during paid hours, and any adverse effects dealt with as with any other work related adverse effects.

But unlike work related injuries, we simply do not know when adverse effect will manifest. If, for example, an employee leaves a company shortly after submitting to vaccination mandated by her employer for continued employment, at most the insurance coverage will extend for a month after termination.