Do they make a FLat RG6 Cable?

Hmm... not that I know of. I think that would defeat the whole shielding feature of RG6 cable, but I could be wrong. I know that the old UHF wires used to be flat, but that's definitely not RG6.

Ask your local Radioshack... they might have a "slim" style wire or something along those lines.

Can I ask what you are trying to do?

Old, but good question, yes:
Eagle Aspen Flat RG6 Coaxial Cable