The cmd workaround was exactly what I needed, thanks!
Try going to preferences in Unity, and select the External Tools tab. In the External Script Editor drop-down menu select Browse, and in the open dialog set the file type to All Files. Go to the directory where Visual Studio Code is installed on your computer (“C:\Users\USER_NAME\AppData\Local\Programs\Microsoft VS Code” for me), and there should be a folder called bin. Open it, and then select code.cmd. Unity should automatically set everything up, and VSCode should now work again.
This is on Windows, I’m not sure how different it would be on Mac or Linux.