What kind of shoes can you use with Garmin Vector pedals? I was looking at these Garmin Vector pedal power meters, and the site says they are "Look Kéo Compatible". Can you put different cleats in them for other brands of shoes? Would they work with regular tennis shoes?
You don't need to buy Look brand shoes to mount Look cleats on them. You do need shoes that have mounting holes that fit Look cleats. Look-compatible mounting holes are the dominant standard for road cycling shoes so almost any road cycling shoe will work (but dominant doesn't mean universal so you should check). 

As for the cleats themselves, there are many third-party cleats that claim to be "Look compatible." Many are fine; there may be some that are such cheap knock-offs that they fit poorly. 

Tennis shoes won't work. The Vector pedals rely on a solid connection between your foot and the pedal so it can accurately measure the forces being transferred. A sloppy connection will result in sloppy measurement.