This is a rather speculative situation. I don't know of any employer who has such mandate for employees at this time, although there could be one.
For there to be any question of liability, there would have to be harm, and reasonable evidence that the harm was caused by the vaccination. If we assume that, then:
The question assumes "There is no specific liability waiver signed by the employee/patient" but this is very unrealistic. I have, for example, hd employer-provided flu vaccinations in multiple years provided by (but not mandated by) multiple employers. There has always been a statement for me to sign that I know and accept the risks of such a procedure. I doubt that any employer would omit to have such a waiver as part of the process. Te effect of such waivers varies, and sometimes they are held to be ineffective as a matter of law. That the employee is required to accept the vaccination as a condition of employment would probably weaken the waiver. That would vary by state.
If the employers are following a medically recommended process, and if known risks are disclosed to the employee/patient, I doubt that any liability for an adverse result would be imposed on the employer, although one never knows what the outcome of a lawsuit might be. I know of no case in which such liability was at issue. if the employer concealed information about risks, or deceived the employee concerning risks, that might well lead to liability.
If the medical provider (physician, nurse, or technician) followed accepted medical practice, was not negligent, took normal case, and made sure the patient was aware of known risks, liability would be unlikely to be imposed o the provider. Again, however, lawsuits are not predictable in such matters. If a special law provided protection to such providers, this would make liability much less likely.
Any liability imposed on the vaccine producer (pharmaceutical company) would not depend on the situation of an employer mandate, but would be the same as for any vaccine recipient. To impose such liability would require evidence that the company has neglected normal standards, and been negligent. It would also require evidence that the company's errors were the proximate cause of the harm done. If a special law was passed protecting produces, it would probably ;prevent any such suits from going forward. Such special exemptions have been made by law in the past.
In short, on the assumptions in the question above. liability is possible but not very likely to rest on the provider or the producer, and very unlikely to rest on the employer.
Addition: It seems from comments that some employers, such as hospitals and the military, already require vaccinations for some diseases. This seems reasonable to me, and does not change my answer otehrwise.