Is it possible to use a USB camera with Arduino?
An Arduino is not a powerful processor and can not do much image processing.  I doubt that there is a USB camera that could be connected to the Arduino.

There are a range of OVO cameras that can connect to the Arduino, but I don't think you will be able to record live video on an Arduino with these, where would you store the file for starters?

There is a PIXY camera system, this has its own microprocessor and interfaces with an Arduino.  It can record video and do object recognition.
http://charmedlabs.com/default/pixy-cmucam5/

Hope that helps.