Is there a term for routers that can share multiple USB devices, including a printer and two hard drives, over a network and maybe even over the Web?
Or can any USB-enabled router generally do this with a hub?
They’re sometimes called Network USB Hubs, or Network USB hosts - or you might find the functionality built into devices described as router with USB host, or you might find them described as comparable to ‘pogoplug’ - one of the popular devices for this function.
The problem you may have is compatibility with the USB devices you want to plug in. USB storage devices can be worked by pretty generic drivers, but printers are a bit more varied and you might need to check if your model is supported. (The device needs to have the right driver to be able to talk to whatever USB peripheral you plug into it).