Download the client for the server you want to play on. I wanted vanilla wow so I needed the 1.12.1 client. A quick google search will give you several results you can download
Unzip the client in the directory you want it to be “installed” in
Go through steam and add the wow.exe in the folder you unzipped as a non-steam game, and force compatibility to the latest proton version available.
In the folder where your wow.exe is there should be a realmlist.wtf file. Open it, delete its contents and replace it with the link your server gives you. For example I registered on everlook.org so my link points to that.
It should run now. Check OP, i edited with a link to a video of a guy setting it up on his steamdeck. Its what I followed.