does climbing work your chest?

Climbing doesn't really target your chest muscles that much. There are situations in which you will need to push your weight up with your hands, which will utilize your chest muscles, but not really to the point where it will strengthen them that much.

It's best to supplement your climbing with chest targeted exercises, such as the bench press, so that when you get to climbing moves that you need to push your body up, you will have the strength to do it.