Yeah, upgrading the forum would definitely be a nicer option; we just need access to the server/database.
The admins don't have this? That's a bit messy.
This is actually exactly like the case with Lemmings Forums in that regard (except that the existing setup was starting to fall apart - little capacity to deal with spambots aside from deleting their posts; if the site went down our only option was to wait and hope it came back up; etc). Of the three admins, two had gone completely AWOL, and the third didn't have server / database access.
In the end, we did go with the move to a new server - in this case it was a fairly small community, making it easier to get everyone on board (and we were at least able to disable posting and put notification topics up) - and I rigged up some very kludgy but ultimately functional code to process HTML backups of all the posts, and convert them into SQL files to import to the new site's database. Needless to say, I've ensured with the new site that all admins have server and database access (even if they don't fully know how to use it - you can ask for help if needed on how to use
server access, but you're up garg creek without a paddle if no one who can give
you that access is around), and even the moderators at least have access to a full backup of the site's files and database (minus the database password). To minimize the need to remember a URL, the new site was "lemmingsforums.net" (the old one being "lemmingsforums.com"). There are limitations without database access - for example, there's no way to know who's voted in a poll, so polls either have to be imported read-only, not imported at all, or the votes reset to zero - but ultimately, given the circumstances I think it went well. We also had access to a very old database backup which allowed us to at least import some
of the user accounts, although many people did need to re-register on the new site; but that's arguably a minor issue.
A similar process would likely be plausibe here (though I'd have to look at the HTML source of a page to work out how exactly a similar importer could be made). There's a much larger community, but at a quick glance it wouldn't seem we're dealing with too much more in terms of number of posts (and any difference would be negated when we consider that in addition to the at-the-time current iteration of Lemmings Forums, we also imported posts from two older iterations for which backups existed).
Of course, given the need to (as much as possible) shut down the existing site, the staff of the current site would need to be on-board with any such idea; otherwise indeed, a split community is likely to result (either that or the new site fails altogether); and at least one admin would need to be available to assist with it (or give admin powers to a user who's conducting the move). Thus, I would not recommend trying such an option without having at least the staff - and ideally, a decent portion of the community - on board with it.