Is it possible to use this mod on a multiplayer server? (as a client)
It's not possible to use most normal mods like this one as Client-Side Mods (CSMs). You should search the client-side modding forum if you want specific CSMs, for instance this one: chatlog. However be aware that due to privacy law in some countries as well as server rules, you may not be allowed to record chat as a client. You should probably ask for consent before recording. The server operator can record chat especially for moderation purposes but a more private server admin will usually limit the amount of chat stored to a period of time like 7 days.