You should be able to connect a USB webcam to the Galileo USB Host port (you will need a UAB OTG adapter) (like this one http://www.amazon.com/gp/product/B006B2VJBU/ref=oh_details_o00_s00_i00?ie=UTF8&psc=1)
The Webcam might need to have drivers installed, so finding one that has Linux support would be good.
Then using OpenCV (which other people have been working with) you should be able to capture a picture from the webcam to a file.
I would then use winscp to transfer the file from the Galileo to the host computer. But that is just me. There might be a better way of doing this, maybe even using OpenCV?