Licenses issued under this article shall be renewed annually on or before June 1 of each year or at the date of engaging such business and shall expire on May 31 of each year; and license fees shall be for the full amount of the tax prescribed, regardless of the date such business is begun. Upon the expiration of a license, it shall be unlawful for any person, firm, or corporation to continue such business until a new license for the privilege of engaging in such business is applied for and obtained.