PDF reader on iOS with autosync
iOS apps can't run in the background constantly to do this task.

Dropbox can do this if you're willing to open the app whilst on network. Add the PDFs to a folder on your Dropbox, then in the app, choose Make Available Offline for that folder.

In the future, you can add files to that folder on your computer, then simply open the app on your iOS device to load the PDFs.