Except as otherwise provided in subsection (b) of this section, a person shall be licensed by the Department as a child care institution before the person may operate an institution for the care, custody, or control of a child alleged to be or adjudicated delinquent or in need of supervision. This section does not apply to: an institution or facility operated by a unit of the State or a political subdivision; or a child care home that has a license under this subtitle or § 5-508 of the Family Law Article.