This information can be gained from the Player Log file!

To retrieve your multiplayer games world seed:

Start up Valheim
From the main menu, click the button labelled Show Player.log hidden away at the bottom left corner of the screen. This will open a folder in Windows Explorer; minimize that and come back to Valheim for now, we'll double back to this in a minute.
Join your multiplayer server as normal. (Start Game -> Start->Join Game, yadda yadda. You know this.)
Once you've loaded into the game world that you want the world seed for, it's time to Alt+Tab back to the Valheim AppData folder that was opened up by that player.log button.
We want to open the text file named Player.log, or, depending on your windows settings, it may show up as simply Player.
In that document, you'll want to Ctrl + F or scroll down and look for the string Initializing world generator seed:. Your seed will appear right after that*. Remember, world seeds are case sensitive!