Does anybody know of a way to get the Windows Update to run for the domain account I login to the PC with, or a way to edit the registry to allow this to happen?
I assume you're not able to run Windows Update on your PC because you're not a member of the computer's local administrators group. This would be the easiest way to resolve the issue. Have your network administrator add you to the Power Users or Administrator groups on the local machine and you'll be able to run updates as normal.
If this is unacceptable, then you could have the administrator do the following:
Update the PDC with the latest Group Policy Management Console and enable the setting to allow non-administrators to receive update notifications for at least you or your OU.
You can then either set the computer to download and apply updates automatically or to notify you of updates or whatever you prefer. This can generally be done on the computer itself, unless that has been blocked by the group policy as well. If so, it would need to be changed in the group policy by the administrator.
Are you an Administrator? If so you can add the user to the local admin group under computer management.