To be tax-exempt under section 501(c)(3) of the Internal Revenue Code, an organization must be organized and operated exclusively for exempt purposes set forth in section 501(c)(3), and none of its earnings may inure to any private shareholder or individual. In addition, it may not be an action organization, i.e., it may not attempt to influence legislation as a substantial part of its activities and it may not participate in any campaign activity for or against political candidates.
“Nepotism" occurs when family members work under the supervision of another family member. It's not illegal but it does present a variety of risks, but it does create potential problems, because of the excess benefits rules and, in particular, the new rules on "disqualified persons," there is an additional incentive to consider the risks of nepotism. However, your situation is really one regarding blocks of voting power, rather than nepotism. As long as these related people are not in a position to benefit one another, there is no nepotism by them all being on the board. Now, you board may want to consider a rule against board members nominating family members to fill vacate board seats.
But to answer your direct question, there is no law that prevents a 501(c)(3) from engaging in nepotism in and of itself. There are tax regulations that prevent particular employees from receiving benefits in disproportion to the value of their service to the organization, but that does not appear to be the situation you are facing.
Please press the GREEN "ACCEPT" button so proper credit is provided for your Answer. Thank you and have a good day. This answer is made available by the lawyer for educational purposes only. It is also to give you general information and a general understanding of the law. This website is not intended to give you specific legal advice. By using or participating in this site you understand that there is no attorney client privilege between you and the attorney responding. This site should not be used as a substitute for competent legal advice from a licensed professional attorney in your state. We attempt to provide quality information, but the law changes frequently and varies from jurisdiction to jurisdiction. The information and materials provided are general in nature, and may not apply to a specific factual or legal circumstance.