Hi AlphaCHarly
I got confused. If you are using the embedded Edison's WiFi, what exactly is the question you have regarding the Ethernet.localIP() function on Edison? If you want to use the embedded Edison's WiFi you should use the WiFi library, but if you want to use the Ethernet library you should attach the Ethernet Shield because the Edison doesn't have an Ethernet port like Galileo has.
Regards,
Diego.