Virtual backgrounds for any videoconferencing or webcam application
The answer is yes. Applications can create a virtual camera, which other applications will have available in exactly the same way they have the built-in camera on a laptop available, or a USB-connected camera for example.

The application would need to read the images from your real camera, insert your virtual background, and then expose it as a virtual camera.

For example see virtual webcams such as: OBS Virtual Camera or Webcamoid.

If you combine the OBS Virtual Camera program with OBS Studio, you can use the green screen (chroma key) features to get the setup you're looking for.