No. System Link requires a separate game disc (or digital license) in each console.
Original Xbox System Link works over LAN (crossover cable or switch). Xbox 360/Xbox One/Series also support System Link for backward-compatible titles (each console needs the game installed/disc).
Bottom line: For a 4-player Halo LAN party, you need 4 Xboxes, 4 copies of Halo, 4 TVs, and a switch.