You can program a Baby Orangutan (and some of our other Orangutan micro controllers) with the Arduino environment. The Programming Orangutans and the 3pi Robot from the Arduino Environment guide can get you started.
I am not familiar with the AVR Wifi library in which you are referring to. Could you please provide a link to its documentation so I can see if it can work with the Baby Orangutan and the Arduino Wifi shield?
That Arduino Wifi shield uses SPI to communicate with an Arduino. Unfortunately, the MOSI line (used in SPI communication) is not accessible easily on the Baby Orangutan. You can access it through one of the ICSP programming pins; however, that might get in the way of programming the Baby Orangutan. You might consider loading the Arduino bootloader onto the Baby Orangutan so you could program it through a serial connection. Please note, the clock frequency on the Baby Orangutan is 20 MHz (Arduino runs at 16MHz) and might require some modification to the bootloader. You might find this thread discussing using the Arduino bootloader at 20 MHz helpful.