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/