Freedom to Travel Act of 2021 This bill prohibits common carriers from denying or refusing to transport a person who is not under lawful arrest except on specified grounds. A common carrier may deny or refuse service if the individual fails or refuses to (1) pay the fare; (2) comply with the rules of the fare; or (3) submit to a legally required search for weapons, explosives, or incendiary devices likely to pose a safety threat. A common carrier may also deny or refuse service pursuant to a court order. The bill sets out penalties for failing to comply with the prohibition, and states are not immune from suits for violations.