I was thinking about making a php page run by a cron job that would go through the database and look up all upcoming expired sites and sending a warning to my clients. However, I only see expiration info inside the config_multisite for each sub-site.

Can you write an extension that would look up upcoming sub-sites about to expire and send emails to the site owner? That would do the job. What is a way I could accomplish this?

For the moment, an email is sent to the owner when the website expired and that somebody connect on the slave site.

We will notice that you would like that we create a special task that could be called from a cron.

I am the site owner but I have never received any emails regarding expired sites. Is there a setting for this somewhere that I may have missed?

Perhaps you have a problem in the setup of your server that is not autorized to send email.

Verify your Joomla "Global Configuration" for the "sender email" and verify also that the owner of the website has a valid email setup in joomla.

Perhaps you also have to check your "anti spam" of you mail box.

I get email from the site when making a new user, it sends out emails, etc. So seems like someting is wrong.

An email is automatically sent when creating a website from the front-end and that you allow automatically generate a new user and/or password to access the website.
This is the only way to let the user know the login information to access the website.