I'm thinking of a solution in which you can use a router (that does not have a current ISP) and use it to transfer data between devices. I know that routers are not normally setting up with the right code to be an end point in a file transfer, so it would require special work. What solution exists that can solve/remedy the problem? Any recommendation would be appreciated.
Yes, a router can be used to connect and transfer data between devices. But actually, what you need is not a router, is the cable which makes the data transfer process simple. For instance, you can connect two of your computers with simple UTP cross cable (a cable with crossed IN and OUT data pair - can be bought with no issues) and you can transfer data like that without any additional device.
If you just get a new router and connect 2 devices like notebook, PC, anything with LAN cable to it, the devices will be assigned IP addresses and you will be able to transfer files between them. If you need to use wireless, you may have to configure the wireless part in the routers interface if a wireless network is not enabled by default.
Note that ISP connection and routing will be irrelevant in this case.
Yes you can, but once you set it up you may find it simpler to throw away the router and just take the Ethernet cables that you used to connect each device to the router and instead plug one of the cables into both devices. Modern Ethernet ports and Cat5e cables are bidirectional and allow two devices directly connected to form their own network. Of course, all the other answers that actually use a router or switch will also work.