Make the VSCode extension compatible with https://vscode.dev/ and https://github.dev/

Is your feature request related to a problem? Please describe.
Developers are not able to use this extension with https://vscode.dev/ and https://github.dev/

Describe the solution you'd like:
Make the VSCode extension compatible with https://vscode.dev/ and https://github.dev/
@allevato @thomasvl @dierksen @reinhillmann @philwo @fweikert
Please consider this as DX improvement and give some priority for it.
Thanks

Describe alternatives you've considered:

No alternatives available.

Additional context:

Here is more info
https://aka.ms/vscode-web-extensions-guide
https://code.visualstudio.com/api/extension-guides/web-extensions

Screenshot of extension showing not compatible with web version.

Start with the VS Code web extensions guide and the linked extension API guide to identify the requirements for vscode.dev and github.dev compatibility. Inspect the extension's current packaging and runtime assumptions, then verify the extension can be used in both web environments without the incompatibility shown in the issue.

Written by the indexing model from the issue text.