As far as I know, the church hasn't made public APIs available for it systems.
Several years ago, there was a push for community involvement in church projects, but that effort was mostly discontinued. As I understand it, there were few active contributors. There might be some projects still available where you can contribute. See
https://tech.churchofjesuschrist.org/projects
You can use the Feedback link, found on many pages, to request such an API. At least your request will go to someone who makes decisions. My personal opinion is that it's not likely the church will expose public APIs. From the understanding of many people in the forums, they have a small staff that is called upon to do a lot of stuff. A public API would bring with it things like documentation, examples, and developer support, which would distract technical people who work for the church from doing the many other things they need to get done. Much of the support for the church is done by volunteer missionaries, and it would probably be more difficult to get volunteer technical missionaries to do developer support. These individuals would need to understand the system architecture, have access to source code and databases, and, ultimately, have access to the developers for when they run into issues they can't handle. Furthermore, that gives volunteer missionaries access to personal information, which doesn't sound like a good idea. I've been involved with developer support at some previous employers and it can be a lot of work. The only way it could make sense is if the church charged for API integration to its systems, which I don't see it doing.