"Web" in the strict sense? Not really, as the "World Wide Web" wasn't invented until ~1990, and the protocol and software weren't made public until 1993[1].
So there weren't a lot of HTTP/HTML based "web" sites between 1991-1994, and probably even fewer "forum" type sites.
BUT... in a more colloquial sense of how "web" might be used retrospectively, as sort of a loose synonym for "online", then absolutely. Going back well into the 1980's, and then into the 1990's, we had all sorts of online services, including discussion forums. There were the big services like Compuserve, The Well, AOL, Delphi, GEnie, Prodigy, etc. And then there were individual small dial-up BBS's[2] by the bushel-load.
At their peak there were many thousands of them in the US alone. There were entire magazines printed and distributed that were effectively nothing but lists of dial-up numbers for BBS's, and magazines like Computer Shopper had big sections dedicated to listing those as well.
And then there was, as you mention, Usenet.
So yeah, even back in 1991, there were plenty of forums that might not have strictly been part of "the web", but that someone in contemporary times might casually describe as a "web forum".
I have a few copy/paste text files from AOL forums from the 90s, of which sadly the other 99.99% was lost unless AOL (or the clients to whom they subcontracted maintenance of forums at individual keywords) still has them on a tape drive somewhere.
I remember printing out USENET posts on fan-fold perforated paper that had green and white stipes circa 1990. I think I got rid of all of them when I moved to Germany in 1999 and really pared down my possessions.
I should really look and see if I have anything interesting on my floppies that never made it to my hard disks. Unfortunately I think I have very little left from BBS forums. I definitely lost some things in hard disk crashes over the years with incomplete backups.
I would assume so! I remember playing on Battle.net with StarCraft back in 97 and that had messaging integrated it. It basically had its own little IRC going. 25 friend limit!
the well has been around since the mid 80s. It was a bbs and then a few other weird things until it became a 'somewhat standard' web forum in the mid 90s -- but if I remember right when it transitioned from BBS there was an effort to move the conversations over to the web forum.
delphi forums too, which similar to the well has been through a few incarnations.
BUT... in a more colloquial sense of how "web" might be used retrospectively, as sort of a loose synonym for "online", then absolutely. Going back well into the 1980's, and then into the 1990's, we had all sorts of online services, including discussion forums. There were the big services like Compuserve, The Well, AOL, Delphi, GEnie, Prodigy, etc. And then there were individual small dial-up BBS's[2] by the bushel-load.
At their peak there were many thousands of them in the US alone. There were entire magazines printed and distributed that were effectively nothing but lists of dial-up numbers for BBS's, and magazines like Computer Shopper had big sections dedicated to listing those as well.
And then there was, as you mention, Usenet.
So yeah, even back in 1991, there were plenty of forums that might not have strictly been part of "the web", but that someone in contemporary times might casually describe as a "web forum".
[1]: https://en.wikipedia.org/wiki/World_Wide_Web
[2]: https://en.wikipedia.org/wiki/Bulletin_board_system