I've never used plex so this might be completely out of the ordinary for even more normal plex use cases, but is there a web client?

I use Android, iPadOS, Fedora Linux, MacOS and possibly in the future Windows (my last job required it, my next job might too) and unless the standard plex app supports audiobooks I doubt I'll find much consistency using native apps for each of those operating systems, if they even exist on all of them

Yes, they have a web client, but their audiobook support is not great. Plex is primarily geared towards music (albums/tracks), and the podcast support is built on top of it by independent clients. I recommend Prologue for iOS and Chronicle for Android.

I've heard good things about audiobookshelf [1] as a cross-platform alternative, but it is pretty early in development.

[1] https://www.audiobookshelf.org/

Seconding Prologue. I have successfully listened to many audiobooks with it on Plex

I'd never thought to use Plex with audiobooks before but this post and guide got me started. Now I've got all my [Amazon audiobook service] books saved safely for when someone inevitably tries to pull the plug on it. And streaming through Prologue is a way better experience than the commercial option