At the time I write this answer, this is not possible. There are two problems on Windows, and one problem on Mac and Linux:
VS Code does not provide a default path to the file dialog 1. It does remember the last folder that you opened a file in, but that path cannot be used as a default because it is overwritten constantly.
On Windows only, Electron ignores the default path when creating a file dialog if the default path is a directory 2.
An extension also cannot solve this, because extensions are not allowed to modify the File menu 3.
I think the best option at this point is to pin a folder to the Quick Access area in Windows Explorer, as suggested in a comment, or to put an actual shortcut in the user profile folder.
Workspaces and File > Open Recent may also be helpful if you often open the same folders.