This is my first post so let me know if there is anything missing...
Based on this thread dedicated to the Galaxy S III, I managed to build the kernel module for the USB Ethernet adaptator for the Xperia Z tablet.
It was not that easy for me, so if other are interested, here is the precompiled file for the stock ROM 10.3.1.C.0.136 (4.2.2).
(I also built it for the Xperia Z ROM 10.3.1.A.2.67, feel free to ask for it)
For any instructions how to use this file, please see the thread linked below in the Galaxy S II forum, it works the same for the Xperia Tablet Z.