Two parts to the question here.
I ... am wondering if it would be OK to ask my employer if I would be
able to divulge that I worked on client A, B, and C's websites.
In some cases employers don't want you to divulge who their customers are. In other cases they don't care. You have ask to find out.
One place they generally don't want you to discuss the list of customers is when you are about to join a competitor. They don't want you to walk out the door armed with information that will help a rival company poach their customers.
the second part of the question:
so that I could land a job at the university.
That means that you want the list not just to add to your LinkedIn page, or to add to your resume; but so you can either quit or take on freelance work that is in competition with them. I know you said you won't quit, but they have no idea if that will end up being a true statement.
First you will have to find out if you can work this second job. Some employers allow it, some don't. You should look at any employee handbooks or other documents HR has. Concerns usually revolve around conflict for time and inability to focus on their customers. They also worry about employees using company resources for non-customers.
Once you know the company rules, then you can ask if you can use the customers list for references.