When working with designers I used to use file locking in SVN but that doesn’t seem to exist in Git though it is on it’s way. This is better than file locking. With a easy to use GUI app I don’t think it will be a huge hurdle to get designers on board — basic Git usage is not that complicated.
How does the app deal with merge conflicts, will the user have to choose between different lines of the JSON file? Or do you tell designers not to work on the same artboard at the same time?