They have a really standard lifecycle: They get built. They work for a small department. If it stops there, it just keeps running until the person who built it moves on, then it gets trashed and replaced with something else. Or, if it grows beyond the department, that person's life starts to get taken up in maintaining it, it fails to scale, and IT takes it over (or replaces it).
The question is not does it happen. It happens so often that there is even a term for it: "Shadow IT". The question is not even whether or not you can launch a product to stabilize the lifecycle. The real question is - should you? The ability to create an app doesn't grant the knowledge of how to make it secure and fit within regulatory bounds for your industry. It is quite possible (and happens often) for someone to make an app for their department that works great and everyone loves, but exposes the company to real regulatory and legal risks.
That is the real reason IT shops have someone to watch over such apps. As you are clearly asking this question in the spirit of self-promotion, you really need to show that you understand the real problems and risks with departmental apps. "Do people build them?" makes you sound like you are completely new to how corporate IT works, which instills no confidence that you have built a viable product.