I am having this same issue as OP described, just started recently.
tl;dr: If your GPU drivers are out of date, try updating them. Below is what I had to do in my setup/situation, I did a full clean removal and install of nvidia drivers and this resolved my issue but may or may NOT resolve OP or others issue.
I launch VRChat via steamVR, have a session, and when I exit VRChat either through in-game menu or steamVR “Exit Game” for VRChat, Steam still reports VRChat as running. There are no processes that I can see in Task manager relating to VRChat.
When attempting to exit the “Running” VRChat from Steam client, it changes to “Stopping” which it stays at indefinitely.
I am also able to replicate this behavior with VRChat by launching in Desktop mode. After exiting VRChat from in game menu, Steam client reports it as running still, and only exiting Steam (using Steams menu to exit Steam, not force killing Steam) and starting it back up does it return to normal for VRChat’s running status (green Play button).
Validating files for VRChat did not resolve the issue.
My Steam client was opted into the beta, I switched back to no beta, and the issue persists.
VRChat is non-beta and up to date.
I am on windows 11 and it is up to date.
My nvidia drivers, however, were still on 566.xx, as there were issues with newer ones in the recent past. I decided to attempt to update them. The update failed, claiming a missing .dll file. I used DDU booted in safe mode to cleanly remove nvidia software. I then downloaded the latest game ready driver and installed it. I chose “Drivers Only” and NOT “Drivers + nvidia App”, then express installation.
After this and a clean reboot, I was able to start VRChat in desktop mode and then quit and Steam client successfully reported VRChat as stopped (changing back to the green “Play” button). I was also able to replicate this success launching VRChat via steamVR.
e: grammar edits