They do. It's called online chess. 
please insert game type
lol:) yea 1 year per move or something....
One of my new years resolutions is to possibly get around to making my moves sometime this year... but some of those games still require a bit more thought and careful deliberation. Until then... *gives self some more vacation time*
(just kidding- we don't really give ourselves more vacation. Or do we?!?
)
An-Aoshak,
It looks like you started a game of e-chess three days ago. If you see this, is that the game type you were looking for? Sometimes it can be a while before the opponent logs in to make their move, so live chess is a good option for in the meantime while you wait.
Why chess.com dont have game type like time per move game?