Too bad. No error messages - its like the Webcam simply doesn’t exist.
I am a bit surprised you needed root permissions to run ‘dmesg’ as I don’t need such.
Anyway, I note:
rsp@dell-linux:~> sudo dmesg | grep idVendor
2.853671] usb usb1: New USB device found, idVendor=1d6b, idProduct=0002, bcdDevice= 5.03
2.858403] usb usb2: New USB device found, idVendor=1d6b, idProduct=0003, bcdDevice= 5.03
2.882093] usb usb3: New USB device found, idVendor=1d6b, idProduct=0002, bcdDevice= 5.03
Those (above) above correspond to the built in (?) root hub
3.294023] usb 3-1: New USB device found, idVendor=8087, idProduct=8000, bcdDevice= 0.04
That (above) is the “Intel Corp. Integrated Rate Matching Hub” noted earlier.
3.693409] usb 3-1.3: New USB device found, idVendor=046d, idProduct=c52b, bcdDevice=12.10
That (above) is the Logitech, Inc. Unifying Receiver, noted earlier.
3.891049] usb 3-1.6: New USB device found, idVendor=0cf3, idProduct=0036, bcdDevice= 0.01
...
35.838852] usb 3-1.6: New USB device found, idVendor=0cf3, idProduct=0036, bcdDevice= 0.02
493.185463] usb 3-1.6: New USB device found, idVendor=0cf3, idProduct=0036, bcdDevice= 0.01
500.424054] usb 3-1.6: New USB device found, idVendor=0cf3, idProduct=0036, bcdDevice= 0.02
That (above) is the Qualcomm Atheros Communications AR9462 Bluetooth. I do not know why it is listed 4 times. Its almost like device was loaded and reloaded multiple times (as indicted by the time in far left).
4.078497] usb 3-1.8: New USB device found, idVendor=0bda, idProduct=0129, bcdDevice=39.60
That (above) is the Realtek Semiconductor Corp. RTS5129 Card Reader Controller.
Again - no error messages with the VendorID of the webcam. It appears that the hardware is simply not working (or not detected if working). Given how old this laptop is, and given it has been reported as working by other users with GNU/Linux, I find it unlikely that it is not detected if it is working and enabled on your PC.
Which suggests as was already mentioned, that there could be a hardware issue.
I note on the web a number of youtube videos showing how to change out the webcam in this computer - which is a long complex process … but the fact that there are such videos suggests to me that having this webcam fail is not uncommon, albeit that is my speculation.