Best way I know to do that is to run a photoshop action on an image sequence you export from your video. Also, Illustrator CS2 has a pretty sweet function to turn pixel based images into vectors with customizable amounts of detail… I want to say it’s called live trace. There is actually a nice COW tutorial about it, although he exports his footage from / back into after effects, and does some other fancy stuff with it along the way.
I realize this suggestion does nothing for you if you don’t have either of these programs, so I apologize if that’s the case. If you have one or both, you should tinker around a bit with the two and see a) if you can get a workflow put together to get your project done, and b) if you can match the style your client wants to keep from their old project. Unfortunately, or maybe the opposite, there’s nothing built into FCP that I would trust / be happy with to create an illustrated effect.
Hope that helps,
Zak