How long before asking for a formal offer letter?
I have recently been [informally] offered a position by a university [and] they said they would send me a formal offer letter along with other paperwork &quot;ASAP&quot;. It has been over a week and I haven't received anything.
I'm not sure how long these things should take...

Far longer than a week! The sheer number of personnel involved---e.g., your future manager, the committee that selected you, the head of school, perhaps someone above them, HR, other administrators,...---means the process is slow. Couple that with numerous inefficiencies and you'll quickly appreciate that 3-4 weeks isn't unexpected (and it might even be twice that).

Would it sound rude or desperate to send them an email to inquire
about why I still haven't received anything from them at this point?

It isn't appropriate at this point. Maybe after 2-2.5 or maybe even 3 weeks.

I would very much like to have some closure on my job search as soon as possible

You already have informal acceptance. I wouldn't expect rescinding acceptance to be particular common. (You could always ask a question about that.)