Fast forward to the 20th century, and you find William Friedman forming and running the FSG (the "First Study Group") in 1944-1946, followed (unsurprisingly) by the Second Study Group in 1962-1963. Incidentally, I don't know if there is any connection between the Dr. Robert A. Caldwell who was part of the FSG and Ian Caldwell (co-author of the Rule of Four), but it would be elegant if so... just a thought! As always, there's more (OK: much more) on the modern history of the VMs on Rene Zandbergen's excellent VMs site.
In the Internet age, mailing lists have picked up this collaborative baton, and run with it in countless new directions. I couldn't see any page that described these lists in an accessible way, which is why I'm posting about them here...
(1) The Voynich Mailing List - the daddy (if not granddaddy) of all modern VMS lists - started life hosted at rand.org (Jim Gillogly's employer at the time) back in 1991, but has relatively recently moved to its own domain, voynich.net Lots of old-timers there, and its old list archives hold an amazing collection of ideas, observations and notions, from a diverse group of contributors: virtually everything you can think about the VMs has been posted there (I search them with grep, but you may prefer other tools). Of late, I think it's fair to say that the discourse there hasn't been quite so good (and it's far from obvious why this should be the case): but it can be an excellent place to start. Recommended.
(2) The Journal of Voynich Studies (2007 list archive here, cumulative file and HTML archive here) is a newish mailing list, run by Berj N. Ensanian and Greg Stachowski. This set out to be more openly academic than the main voynich.net mailing list, but it unfortunately (I think it's fair to say) still falls well short of the high academic ideals to which it initially aspired. The reason for this is obvious: there are, to my knowledge, no 'true' academics (in the 'scarily and unchallengeably erudite academic' sense, how I imagine Panofsky or Grafton) out there studying the VMs. Really, the basic art history still hasn't been done: Voynichology remains an amateur science at best.
(3) Quiet-ish groups, such as thevoynichmanuscript Yahoo Group may well be good in a different way. I'm not a member there, but it might be just right for you.
(4) Transient mailing lists (such as GC's Strong Solution group from 2006, or the Voynich Forum) come and go, ebb and flow: this is inevitable on the Internet.
(5) Non-English mailing lists have started to appear in recent years: I'm thinking in particular of the German Yahoo Group voynich.de. I have a vague memory of Jan Hurych starting up a Czech-language mailing list at one point, though I don't know if that is still going: his current Voynich blog is here. Similarly, there was a Spanish-language MS408 list back in 2003, and I half-remember a French-language mailing list flickering into existence too.
Doubtless there are/were/will be more: but that should be enough to get you started. ;-)