I run into the same problem and fixed it by using SSH ControlMaster for Single Singe-On.
If you connect to the relevant machine using <user_name>@<host_name> you can simply add the following to your ~/.ssh/config:
Host <nickname>
User <user_name>
HostName <host_name>
ControlMaster auto
ControlPath ~/.ssh/%r@%h:%p
If you now open up a new session in your terminal running ssh <nickname> you will be asked for the password and the 2FA key. But all other subsequent ssh sessions (including svn, rsync, etc. that run over ssh) will piggyback off the initial connection without the need for re-authentication.
You can now in VS Code, select Remote-SSH: Connect to Host... from the Command Palette (F1, ⇧⌘P) and simply enter: <nickname> and ControlMaster will automatically connect you using the already standing connection.
In my case, the nickname that I choose is "ody". After connecting VSC's Remote Development extension shows the following in the bottom left corner and lets me browse the files on the remote machine.
From the FASCR:
Note that all subsequent connections are dependent on the initial connection — if you exit or kill the initial connection all other ones die, too. This can obviously be annoying if it happens accidentally. It’s easily avoided by setting up a master connection in the background:
ssh -CX -o ServerAliveInterval=30 -fN <nickname>
The -fN make it go into the background and sit idle, after authenticating. (C for compression,Y for X forwarding, and o ServerAliveInterval=30 to prevent dropped connections have nothing to do with the ControlMaster but are almost always helpful.)
Note that all port forwarding, including X display forwarding, must be setup by the initial connection and cannot be changed.
