Is it possible to connect Xbox One S controller to Windows 7 via Bluetooth?

Yes, you can, however you need a specific model of controller. In the image below, take note of the top-front of the controllers. Xbox One controllers that are Bluetooth compatible have a solid face plate with no gaps, whereas non-Bluetooth models have diagonal splits along the shoulder buttons:

You will also need to update the controllers firmware. To do this, either connect the controller to your Xbox One while connected to the Internet, or connect it to your PC with a micro USB cord. See this link for updating the firmware through a PC.

To pair the device, hold the sync button on the top-left of the controller while you search for new Bluetooth devices on your PC. It should appear as "Xbox One wireless controller for Windows".

If you still have issues connecting your controller, you can also use the Moga Serial to Windows Interface tool to use the controller as an XInput HID.

I have used all of these methods successfully on Windows 7, 8, and 10.