The short answer to this question is that federal law allows private employers to require vaccinations as a condition of employment. It is anticipated that the law will not treat the adoption of a COVID-19 vaccination policy any differently. Indeed, the EEOC has now opined that the COVID-19 vaccination itself is not a medical exam.
This means employers can require the vaccination as a condition of employment, although employers need to be mindful of employees claiming an exemption from the vaccination policy based on health-related or religious reasons. Employers who will be administering the vaccination to their employees will also need to clear another hurdle. Consequently, the rule does come with exceptions.