How to perform a self-update on a local web system?

Question

I developed a web system (in CodeIgniter) for a client and it runs on a local machine inside his office. Whenever I need to update the system, I perform the following steps:

1. Access to server machine via TeamViewer;
2. I open CMDER;
3. Access my account on Bitbucket to give a git pull;
4. I run the necessary migrations;

Thinking about these desktop systems that automatically look for a new version and upgrade, is it possible to do a routine in this web system to do the same thing? Search Bitbucket (or another web server) for a new version and then do the update?

Thanks for the suggestions.