The best (and cheapest) way I can see to accomplish this would be to hook the scanner to a computer that is on most of the time (desktop) and then share the folder it scans to. Then just map the share as a persistent network drive on the other computers, that way the other PC's will reconnect and map the drive when they turn on.
I don't know what operating system you have so I will do this for Windows 7 which is close enough for 8 and vista as well.
So if you have a folder you scan to, C:\scans for example, you can right click> properties then click sharing (not advanced sharing as the standard sharing will set NTFS permissions for you as well) type everyone and click add. You should now have sharing and NTFS permission set to read for everyone, this shouldn't be an issue if your network is behind NAT. However you can also set it to specific user then specify the credentials when you map the drive.
Now on the other PC's (again assuming windows) right click "My computer" and click "Map Network Drive", pick a letter further down the alphabet so it doesn't interfere with any PNP devices. For folder type
\\yourpcname\folder the PC name can be found under properties after you right click my computer and the folder name is the same name it has on the host PC so in our example: "scans". The first checkbox "Reconnect at logon" will make sure this drive remaps any time this PC logs on and the host is also online, the other checkbox is for using credentials if you don't want your share readable by everyone.