I am aware of a western US State Department of Corrections that local merchants will ONLY deal with on a cash basis.
How's that for irony?
There's too much going on (including phishing, which is not impossible on the phone) to give out sensitive information over the phone without verification of the source. And it's unlikely that a uni or gov agency would operate without a written alternative for submitting sensitive information.
That's why you initiate the call to the business office to gain clarity about the school's policies. The number is publicly accessible.
It is not unlikely that the uni has a form (in quadruplicate, or more) that needs to be completed and sent in. It is govt. after all...