Not surprisingly, the American Lunar Society is exclusively committed to the study of the Moon, and the Association of Lunar & Planetary Observers (ALPO) is substantially so. Both web sites are well worth visiting.
The Astronomical League has a Lunar Club which awards a certificate to any member who follows its excellent 100-point observation program.
In the UK, both the Society for Popular Astronomy and British Astronomical Association run dedicated lunar sections.
The The Moon Society is an international non-profit educational and scientific foundation working to further scientific study and development of the moon. It is closely associated with the Artemis Society International, essentially a private venture to establish a permanent, self-supporting community on the Moon.
Moonlink is an education program designed to tie in with NASA's Lunar Prospector mission.
Organisations with more specific aims include the International Occultation Timing Association (IOTA) and the mainly Islamic Committee For Crescent Observation.