Texas governor Greg Abbott issued an executive order on Monday banning all state entities from enforcing vaccine mandates. This ban will include private employers. This allows anyone in the state to object to receiving the COVID-19 vaccine “for any reason of personal conscience, based on religious belief, or for medical reasons.” He has asked the Texas state legislature to pass a law with the same effect.
See "Texas governor bans vaccine mandates by any employer in the state", Paul LeBlanc, CNN, October 12, 2021