There are a number of plant societies around the world that focus completely or partially on bulbs. Some focus on a particular region, or a single genus, while others are more generalized garden societies but pay a lot of attention to bulbs.
Many of these associations have seed and bulb exchanges. Be aware that the seed exchanges, especially the ones run by the larger plant societies, sometimes produce complaints. Many people love them, but some report that they often receive misidentified seeds.
Generalist Bulb Organizations
Pacific Bulb Society
Despite the word "Pacific," this organization is open to anyone, and covers bulbs from all over the world. The PBS is creator of this website, so it would be just plain rude not to join. Besides, members get to participate in the society's extensive seed and bulb exchange program, which includes many rarities that are hard to find anywhere else.
Indigenous Bulb Association of South Africa
Dedicated to South African bulbs. Benefits of membership include a monthly meeting, email reports, occasional several day symposiums with speakers and field trips, and a seed exchange program for South African members. This group no longer has an active website, but there is an archive from 2017 here that lists their vision and mission and has links for early history and some of their publications.
International Bulb Society
Exactly what it says, from 1933 to 2013 this was a society devoted to bulbs in general (although it's probably fair to say that many of its members had a special enthusiasm for Amaryllids). The website www.bulbsociety.org is no longer active as of January 2014.
South African Bulb Group
Based in the UK, but membership is open to anyone. Focuses on bulbs from South Africa.
Genus-Specific Bulb Organizations
American Daffodil Society
Narcissus shows, commercial sources, and extensive information on species and hybrids.
American Iris Society
A very active organization, with many local chapters in the US.
Aril Society International
Focused on aril irises; members can participate in rhizome and seed sales. Distributes mostly hybrids, but also offers some species, which are hard to obtain anywhere else.
Clivia Society
Based in South Africa, it hosts a lot of Clivia information online. This organization partners with regional Clivia groups such as the North American Clivia Society.
The Cyclamen Society
Publishes a twice-yearly journal and runs a seed exchange.
North American Lily Society
Focuses on Lilium species and has an annual lily seed exchange.
The Society for Louisiana Irises
Species Iris Group of North America
As you'd expect, the focus here is on species Irises. Has an active annual seed exchange, and despite the name is open to members from anywhere in the world.
Garden Societies That Pay Attention to Bulbs
Alpine Garden Society
The leading rock gardening association in the UK. Has a very active Fritillaria group.
Mediterranean Garden Society
For growers of Mediterranean-climate plants around the world.
North American Rock Garden Society
The leading rock gardening association in the US and Canada.
Scottish Rock Garden Club
Although this is a generalist alpine plants organization, it puts a heavy emphasis on bulbs. Its Bulb Log is an outstanding diary by a bulb enthusiast.