BINGO

Here's the statute: 849.0931Bingo authorized; conditions for conduct; permitted uses of proceeds; limitations.—

This paragraph covers HOAs:

(4)The right of a condominium association, a cooperative association, a homeowners’ association as defined in s. 720.301, a mobile home owners’ association, a group of residents of a mobile home park as defined in chapter 723, or a group of residents of a mobile home park or recreational vehicle park as defined in chapter 513 to conduct bingo is conditioned upon the return of the net proceeds from such games to players in the form of prizes after having deducted the actual business expenses for such games for articles designed for and essential to the operation, conduct, and playing of bingo. Any net proceeds remaining after paying prizes may be donated by the association to a charitable, nonprofit, or veterans’ organization which is exempt from federal income tax under the provisions of s. 501(c) of the Internal Revenue Code to be used in such recipient organization’s charitable, civic, community, benevolent, religious, or scholastic works or similar activities or, in the alternative, such remaining proceeds shall be used as specified in subsection (3).

It appears that no state license is needed. But maybe your county has a requirement.

I can tell you that our HOA in Florida regularly runs Bingo and we do not have a license and have never had any issues.

Please note that HOAs and COAs are not for profit (not a nonprofit).
Therefore, any revenue from the games would be considered taxable income to the HOA.