Hi rossyriot and welcome.
Unfortunately, I can't find detailed specifications for your model - these are missing from the Vaio website - so I can only talk from experience of other models.
I assume that if your model has a 'WEB' button, it supports Quick Web Access.
Quick Web Access only works if your Vaio if fully turned off - if it is turned on, then it does as you say and launches the default web browser in Windows.
Make sure your Vaio is not in Sleep or Hibernate mode but fully closed down and press the 'WEB' key. This should open an initial setup screen which you need to complete. When setup is completed, the WEB button will open the QWA Browser when pressed as long as the Vaio is shut down.
If it goes to Windows and opens the default browser when it is shut down, you can try reinstalling QWA or give Vaio Support a call.
