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/
@hbenl @rugk @Misiur @jonfen @ilikesortedarrays @jnachtigall
Please consider this as DX improvement and give some priority for it.
Thanks

Describe alternatives you've considered:
No alternatives available.

Additional context:
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 linked VS Code web-extensions guide, then inspect the extension's configuration and entry points for assumptions that prevent use in vscode.dev and github.dev. Done means the extension is recognized as compatible and can be used in both web environments; the issue does not name a specific file or test to run.

Written by the indexing model from the issue text.