Unless something has changed in the last 3 years, which knowing the internal systems of the LDS Church it is highly unlikely anything has changed, there will not be a JSON feed for the Book of Mormon anytime soon. They store all of the scriptures in proprietary XML formats inside of a large XML data store called MarkLogic. Although they could transform them into JSON, I just don't see them making that a priority.

However, I will also note that the Book of Mormon itself is no longer under copyright. The footnotes, indexes and chapter summaries are copyright protected but the book itself is not. So, really, it wouldn't be particularly hard to make a Book of Mormon rest api.