I'm going to go out on a limb and say that C3 could do this but. You'll be writing a lot of custom JS code to achieve it.
C3 projects have 3 major ways of connecting to other systems.
- Multiplayer Plugin
With Ajax you're sending requests to a server and getting responses back.
With Websockets, you're connected to a server and sending data back and forth.
With Multiplayer, you're communicating with a signalling server to find matches and then one player becomes the host/server for the other players to join.
The Multiplayer Plugin would work well on a local network as you're looking to do as latency would be low. However, you'll still need the internet to contact a signalling server and match the players to each other. Once they're matched, the app that's hosting the game takes over.
Trying to create a multiplayer system that would only function on a local network without the signalling server would require a LOT of work as you'd be writing code defining ports, communications protocols and handling the network traffic back and forth to manage players. Essentially, writing your own plugin.
There may be another way to do this and someone may give you a better answer, but I'm not aware of it.