I have a cPanel account that is over quota, and his websites are still online. Is this normal?
Yes, this is normal behavior. Reaching the disk quota of the account will not cause websites to go offline.
We do have a feature request to implement behavior on accounts that have exceeded their disk quota, similar to the current behavior when an account has exceeded its allotted bandwidth. An error page would be displayed instead of the website. That feature request is located here, for your reference:
We strongly encourage you to vote on this feature request if it described the functionality that you desire. The more votes a feature request receives, the more likely our developers will implement it.