Sound like a great idea! Just so you know, you are not the only one who is thinking in this direction. Here's a sample of a succesfull app:
https://mobilunity.com/blog/outsourcing-company-mobilunity-launches-time-management-android-app-time...
It actually makes sense and pays back. Though you need to get some help from professional engineers in programming like these guys.
By the way, everybody has smartphones these days. It's totally ok to expect your employee to have one. Personally, I'd like to have an app on my phone that would help me to be more productive at work.
