If the site is not using some form of authentication, you can use your browser to discover the URL behind the button. Press F12 on firefox or chrome or edge to open the developer pane and watch the Network tab. Click the CSV download button and study what happened...you are looking for a GET request and the URL it used. Right-click File->CopyValue->CopyUrl might get what you need. Press F12 again to close the developer pane.
If the site requires authentication (user name, password) you might find Browser Scripting with Python Selenium helpful, but the learning curve will be longer.
Craige