After working without issue, a Package server begins throwing a critical error: 'Package server is configured to publish http(s) codebases, but could not access it's own web site.'

The disk drive that both IIS and the Altiris agent were installed on (C:\:) was out of available space.

IIS will fail to process all requests if the drive it is installed on (and logging to) is full.

Clear space from the drive, by deleting unneeded files and IIS logs.

IIS has no built in method of clearing its own outdated logs. Please see the following article from Microsoft of way to manage the IIS log files:
https://docs.microsoft.com/en-us/iis/manage/provisioning-and-managing-iis/managing-iis-log-file-storage

Note: Manually deleting Software Delivery, Software Library, or Patch Management packages is not supported and will lead to errors being generated in the logs. Policies for these software packages need to be deleted and time needs to be allocated for the configured automatic deletion to occur.