You will probably need your University's "EIN" (their taxpayer ID number, which is somewhat similar to a person's Social Security Number) if you wish to claim an educational tax credit or benefit, or if you received a W-2, or 1099, income statement from them, and you now need to report it on your own income tax return.
Obtaining the EIN should be relatively straightfoward; it's a routine request. Moreover, you have a legal right to know it, if you are claiming a related tax benefit or reporting taxable income, based on your activities at the University. Usually, in cases like these, a simple phone call inquiry, or an in-person visit, to the University's Registrar office, will get you what you need.
The EIN should be 9-digits long, and appear in a format like: XX-YYYYYYY
In the unlikely event that the University is unwilling to provide you with this simple information, and they require a more formal request mechanism, you can fill out and then present them with an IRS Form W-9:
This is an official document used by taxpayers to (legally) request the tax ID number of another person or entity, for a legitimate purpose. This legitimate purpose would include the necessity of known their EIN number, so that you can file your own tax return.
Thank you for asking this question.