Is a marriage legal if a marriage certificate is not issued?

That would depend upon the laws of the state in which the union occurred. However, state laws require that the person who can perform legal marriages must issue a certificate of marriage to the couple after the ceremony has been performed. A copy of the certificate is then made a part of public record. Certain government agencies such as the Social Security Administration require a certificate of marriage before they will change the woman's name on SS records (assuming she wants to take her husband's surname). A marriage license is not acceptable proof that the marriage has taken place.