Let Our Cities And Local businesses Help Employees Achieve Long-Term Health Act or the LOCAL HEALTH Act This bill authorizes private businesses and local governments and agencies to require the use of face masks and take other measures aligned with Centers for Disease Control and Prevention guidelines to prevent the spread of COVID-19, provided that the required measures accommodate individuals with disabilities and include exemptions for sincerely held religious beliefs. The bill also (1) prohibits a state from establishing or enforcing a law that restricts an authorized measure, and (2) provides immunity to private businesses and local government entities from federal or state claims arising from an authorized measure unless the claim involves negligent behavior or willful misconduct.