Yes, there are still people believing that the earth is flat, and they do have an organization, founded by one Samuel Shenton in the 1950s. Daniel Shenton is the current president (and he is, curiously, apparently not even related to Samuel). According to Shenton, it is not gravity that pins us to the ground but the rapid upward motion of a disc-shaped planet (that Newtonian thing about constant motion seems to have escaped him), and you can indeed fall over the edges. He also uses a GPS when riding his motorcycle, apparently.
Of course, to accept a flat earth cosmology you have to accept some conspiracies, and Shenton is happy to grant for instance that the moonlandings were faked. Apparently the idea that motivates him is Zeteticism, which according to Shenton “emphasises experience and reason over the ‘trusting acceptance of dogma’,” and the earth feels flat to him. Actually, it doesn’t really, but whatever. Apparently the evidence that convinced him was Thomas Dolby’s 1984 album The Flat Earth.
In fact, Shenton comes across as a rather curious case. He has no problem with evolution, and he does think there is good evidence for man-made climate change; he was accordingly deeply offended when Obama compared global warming denialists to flat earthers.
Diagnosis: It is, to be honest, a bit unclear how deeply committed Shenton is to his idea, and it is hard to imagine his society having any lasting, detrimental effect on civilization. Indeed, one may compellingly argue that they provide a good and helpful illustration of how denialism works, and how silly it actually is.