The OED says yep developed as a colloquial alternative to yes in the late 19th century in the US. Used as an adverb and interjection, it is also spelled as yip and yup.

Yes itself is quite old, originating in Old English, and has quite a few variants, including yea, yaas, yah, yas, yeah, yeh, yep and yerse.

This might help with the origin of the word, but unfortunately the OED doesn't say how it became common.

(The OED also points out that yep developed as a natural exclamation to urge on a horse in the late 17th century, but this is a different meaning.)