Unitarian Universalist and related groups
Unitarian Universalists for Polyamory Awareness: an organization for UUs who support ethical and responsible non-monogamy.
Interweave Continental: UU organization for GLBTQ+ concerns.
Religious Institute for Sexual Morality, Justice and Healing: independent organization devoted to promoting sexual healthy religious communities, working closely with the UUA and other progressive religious groups.
BDSM and related groups
Community-Academic Consortium for Research on Alternative Sexualities (CARAS): facilitates interaction between academic researchers and BDSM/kink/fetish, polyamorous and GLBTQ+ communities which they study; the research they help support could be useful for ministers and religious educators.
National Leather Association–International: one of the largest BDSM/kink/fetish organizations, with several chapters and numerous projects and resources.
National Coalition for Sexual Freedom: alliance of BDSM/kink/fetish, polyamory and swing organizations committed to creating a political, legal and social environment in the US that advances equal rights for consenting adults who engage in alternative sexual and relationship expressions; hosts the Kink Aware Professionals list.
Sexual Freedom Legal Defense and Education Fund: helps with legal referrals, information and raising funds for defense efforts on behalf of people involved with alternative sexual expression. (And … run by a Unitarian Universalist!)
Woodhull Freedom Foundation: organization devoted to promoting sexual freedom as a fundamental human right.