Generally speaking, employers have an obligation to provide a safe work environment for their employees. OSHA regulations and workers compensation laws were enacted to make sure that this obligation is met. For that same reason, employers can mandate vaccinations or other other safety measures for their employees. The science and medicine on COVID supports these actions by the employer. The EEOC issued guidance indicating that they do not view vaccination mandates as violations of federal EEO laws, subject to a few limited exceptions. Mainly, consideration for religious beliefs and ADA accommodations. (https://www.eeoc.gov/newsroom/eeoc-issues-updated-covid-19-technical-assistance.) To be clear, this does not mean that unvaccinated persons need to be let into the the workplace, but only that employers needs to engage in the interactive process to determine whether or not there can be alternative arrangements. At least one Court has recognized the legality of an employers decision to require vaccination in Texas, see Bridges v. Houston Methodist Hospital.
With the recent FDA approval of the Pfizer vaccine, one can expect more employers to adopt this policy. However, given the large number of people who have been working remotely over the last year, it would seem that working remotely has established itself as a reasonable accommodation in many more settings than it had pre-COVID.