I would accomplish something fairly close to this by using Image objects.
Your requirement is for 4 options, so I would add 4 separate Images. Each image will use an Expression for it's source, choosing between two embedded images to show On vs Off (for that option out of the 4).
Layout might be in a single column with 4 rows to mimic a typical radio button layout.
The data source for the SSRS report would be a stored procedure. It would return the On/Off state for each option as 4 separate columns - these would drive the image selection expressions above.
For each Image object, I would set the Action to "Go to Report", pick the same report and pass a parameter value indicating which button was chosen.
When the user runs the report and then clicks one of the images, this will fire the stored procedure again. During it's execution I would update a table to store which button was chosen for that row.
The stored procedure needs to execute very quickly e.g. sub 5 seconds, ideally sub 1 second, otherwise the UX will be poor.