The wiz812mj/wiz811mj use the W5100 chip just like the Arduino Ethernet and Ethernet Shield. The Ethernet library just works once you have made the usual connections.
This module, looks really nice and small but does use the W5200 chip. I've seen reference to modification to the library being required, http://blog.wiznet.co.kr/w5200-arduino-ethernet-library/#.T_y6hMXl9f8
. You get more open connections which might be nice. It should be easy enough...I might add one to my next Mouser order.
I've picked up W812MJ modules from Semiconductor store, but I'm not sure they were actually stocking them yet so I waited a while for delivery. Mouser started carrying WIZnet products about the same time as Semiconductor store and has some WIZ820io in stock. I've bought some W811MJ modules from WIZnet directly in the past, http://shop.wiznettechnology.com/
would be another place, but I have no personal experience with them.