What if the window frame and trim on the sill are not in wood?

Looking at the inside of your sill cover it looks like you might be able to pry it off using a chisel. However, check to see if the sill cover is under that side trim of the window or if the sill cover is notched around the side trim. You don't want to accidentally break it because that side trim is in the way. 

Also, if you can get your hands on an off-set Phelps screw driver, that might fit under the sill cover and you wouldn't have to remove it at all. 

Whatever you do, make sure to take your time so as to not to damage the window. Vinyl can crack very easily during a project like this.