I also have a VAIO Laptop with a Motion Eye Camera, which is working very well.
There is one difference between my camera and yours, check this:
The output of lsusb is this:
Bus 005 Device 004: ID 0ac8:c002 Z-Star Microelectronics Corp.
This part is only the one of the camera.
This is the Web page where I downloaded the driver:
Before installing check if your camera is supported!