Arduino - receive Ethernet W5100 signal to multiple arduinos
The W5100 chip connects to the Arduino via an SPI interface. This is a bit-serial interface (but not async serial RS-232). SPI is a master-slave interface: the Arduino is master, and the W5100 is slave.

No, you can't share a single W5100 across multiple Arduinos. The best you can do is to add a fifth Arduino to act as the master to control the W5100, and then connect that Arduino (somehow) to the other four.