Does rock climbing build chest?

Not really. Climbing is more of a pulling exercise for the upper body. This basically leaves your chest and triceps sort of neglected, which is why many climbers do antagonist training such as push-ups, dips, and bench presses.

There are some climbing moves such as mantles that use your triceps, and to some very slight extent, your chest. But for the most part, your chest is not used enough to build it a lot, at least not as much as your other muscles are used, such as your back. Which, again, will cause an imbalance in your muscles if you don't train your chest and triceps specifically. Your posture could be effected by this quite a lot.