Can I set up a LAN network that connects a computer to a bank of firewire drives or does a LAN network only work if a second computer is involved?
Some drive boxes are called NAS or "Network Attached Storage" meaning they have LAN ports built-in, attach directly to the network, and do not need to be attached to a computer in the network to be seen by other stations on the same network. However, basic drive boxes that have only firewire or USB ports will need to be attached to the network, and this requires a computer or network server. The good news is it does not have to be anything special in the way of computing power; any old computer with GigLAN and the proper connectivity ports for your drive boxes will work.
If you need to buy a computer for this purpose, then it might be more cost-effective to sell your old enclosures and replace them with NAS boxes. There are a bunch of NAS options available and they are getting reasonably priced IMO.