Hello Maxwell and welcome to WPSE 🙂
I want to believe that when you talk about:
- an “application” on a site about WordPress, you most likely refer to a “Plugin”.
- “a section of a site, separate from the WP admin” you probably mean “A page or Post in the front-end (site/blog)”
- “the documents themselves could not be linked to or accessed by URL unless logged in” you mean that they are “Password Protected” with a “URL that is NOT traceable”.
If my assumptions are (hopefully) correct then you are looking for a Download Manager. There are a few in the WordPress Database and these are the links to a couple of them.
WP-Filebase Download Manager, very well supported with a sync feature. !!
WordPress Download Manager, never tried it but looks quite simple and very user friendly.
WordPress Download Monitor – we use it and are very happy with it. 🙂
Please have a look and see which one meets your needs.
I hope this helps. Let us know how you do,
marikamitsos