The Seattle Library has 27 locations and a million books, serving a city with 700,000+ people (and a larger metro area - I forget if you need to be a Seattle resident to get a card).

To be able to query materials across the library system, handle reservations and item transfers, and so much more these computers absolutely need to be networked.

Perhaps you meant "don't need to be connected to the Internet", but I would wager a majority of the library's patrons search the library's catalog online rather than coming in person.

In the case of digital loans, moat patrons want to download their digital books onto their personal devices - over the internet, so clearly the library services need to talk to an online service.