Wiring a 7 pin laptop webcam?
Examine pinout of known chips. There seems to be some memory IC (SOIC-8), where it should be easy to determine which is Vcc, and which is GND. This is the first step, to determine which wire is ground, and which is power. The connector actually says that top (grey) wire is 5V. You should take it as solid hint. Trace it to a ceramic bypass cap; the other end of cap will be ground.
If violet-blue form a twisted pair, they could be the data D+ and D-. You should be able to trace them on the PCB. If they are high-speed data, they should stay as parallel traces all the way down to controller chip. 
Then apply +5V power from USB cable, and watch which data wire pulls up. This will be  D+, the other is D-m, and can be connected to "green" and white wires of USB cable.
It is possible that the twisted purple wires are connected to a microphone, so the actual USB data are somewhere else. Then green-yellow might be USB data. Watchig the Pull-up high will tell you which is which.



  Does it even make sense to try to connect it to a USB first


Yes, it makes perfect sense, since it is a USB high-speed webcam. It will work as webcam for PC, and you can make some use of it if it works. However, I am not absolutely sure how do you plan to use a HS USB camera with Arduino, which is a low-end processor and can't have a function of HS USB host.