It is possible, but isn't supported natively by the DIR-320.
The way to go about this would be to get a router that supports dd-wrt third party firmware, has a USB port and has USB support for scanners via libsane within DD-wrt (SANE linux scanner software), and then have a scanner that is supported by libsane.
Then you would need SaneTwain to interact with the scanner across the network.
There are lots of points in this chain that you might have difficulty getting to work. It may be simpler to get a cheap nettop PC, run windows on it and plug the scanner into it. Then "share" the scanner on the network by having people access the Windows box with RDP.