Take a proper close-up photo of your webcam. Use the macro setting on your camera if you have one, Separate the connecting wires apart so they can be easily seen. Focus you camera on the area that is not protected to see if we can see any components that might give a clue.
You might end up having to carefully remove the tin cover to find the proper wire.
In my experience, generally the positive wire is either red or white, the ground is black or green, and signal wires are any other color. There also appears to be a QR code on the camera, you could try scanning that to see if you can find any more information on the camera from the website but that's all I've got.
Is that red wire right next to the Brown wire? That would be a good guess for a power wire. Are there any visible traces or components on the reverse side?
Is it still possible to measure any voltages from the laptop where you disconnected it?
It would be unusual to use light brown or blue for power so I would have to go with red. I understand that making the wrong choice could be disastrous. Unless you can safely get inside that metal shield, I don't have any more suggestions.
I expect you have microphone on that module and I expect mic wires are one next to each other. So if dark brown is GND, light brown and red would be microphone. And light blue (3.3V) positive.
Just my logic....
The first wire (the blue one with the arrow) is the vcc the last wire (orange/light brown) is the GND. The black wire is data+ and the white is data-. The dark brown and red wires are probably the mic wires (I haven't tested).