It's theoretically possible. Any capture card that uses UVC and/or UAC will work in Raspbian. However, this would probably not be very practical due to the Pi having not a whole lot of power and it may struggle with higher resolution video.
An example would be a MiraBox (~$100 on Amazon - I'm sure there are cheaper options however) - which would work on the Pi but HD resolutions may be a challenge to record. A better solution would probably be a portable capture card like an AVerMedia Live Gamer Portable as it would likely work out cheaper and more effective in the end.