Use excellent plug-in KeeAgent which works similarly as integrated Pageant (putty tools) and once you unlock your passwords in KeePass KeeAgent allows you to access your SSH key through Agent.

To allow use of your KeeAgent or Pageant SSH key in Visual Studio Code you would have to use Plink utility which function as an console ssh client supporting Pageant or KeeAgent cached SSH keys.

Unfortunately Visual Studio Code does not support PLink automatically but it supports command-line ssh client. For this reason there is a project ssh2plink which acts as ssh console client but is based on Plink which supports KeeAgent.

You can get it here SSH2Plink

Step 1 - Compiling SSH2Plink

git clone https://github.com/MarkusDeutschmann/ssh2plink
choco install go (if you do not have go language installed and have choco package manager)
./build.cmd

Step 1 - Download builded images (if you want to)

ssh2plink_32bit.exe

ssh2plink_64bit.exe

Step 2 - Copy file for your platform into Putty directory

C:\\Program Files\\PuTTY

Step 3 - Set it up in Remote-SSH plugin

The next level - Windows Hello

The next level for this set-up is to use Windows Hello plug-in for KeePass which allows you to authenticate through your Face (e.g. webcamera Logitech Brio supports it or it is integrated in HP EliteBook) or through your fingerprint or PIN code.