Hello @Laurent.78 and thanks for the details.
on-the-fly- analysis of the source code modified based on the Sonarqube rules (ideally)
This is exactly what SonarLint should provide in connected mode with SonarQube. And yes, SonarLint supports C analysis in VSCode, as long as you provide the compile_commands.json.
In order for our teams to investigate, could you please capture the full logs that SonarLint generates when you open a C file in your project and send them to us (you can see here our to do it).