I would hold out a bit on buying a webcam. We've got one guy that purchased the webcam in Daytona 3 in hopes that it might fix the billboard issues.... if it does then obviously that would be the one to get. (It's about the same price.)
Since a webcam is a hardware device and the games tap into it directly, there isn't an easy way to rotate the image. It might be possible to route it through a software based webcam like splitcam and rotate it via software, but most likely what will happen is the game will detect the real webcam as the primary camera and it wouldn't work anyway.
By far the easiest solution would be to mount the camera on a servo arm and use a cheap avr to rotate it when needed.
As for games that use webcams, I don't know of a lot. I believe there are some in arcades right now, but unless they are made working that won't help you.