I don’t think he can legally demand to see the jab paper because of the Hipaa law. He will or should get sued on this.
IIRC - HIPPA only means a health care professional can't disclose your records to an unauthorized person.
HIPPA does NOT mean that absolutely anyone/everyone can't ask you about a medical condition.
HIPPA will not be a legal winner for someone suing because a restaurant asked your status.