I can't say what procedure you must follow. It depends on the company policy, etc.

Technically, it doesn't require anything at all except running the script. It's policy, customer and technical desires, etc., that make it more complex.

Most of the time, it shouldn't require any down-time to deploy a new proc (or modified version of an existing proc). If it requires any changes to the front-end application, then that might require some downtime, or might not, depending on the application and the change.

The right thing to do is figure out what impact the new proc will have. Will it affect what customers see/do? If so, letting them know is at least polite, and possibly necessary.