It depends on what ability the Azumarill has. I'm not sure what Azu's role in nuzlockes are, but I can gives some of my thoughts.
In my opinion, Sap Sipper is probably the better ability over Thick Fat due to the extra immunity. If you're still in the early game, it can be useful against Ramos and his Grass types, as well as Wulfric's Abomasnow and Diantha's Gourgeist. Thick Fat can also be used to sponge hits from the likes of Wulfric and Malva, although I'm less sure about how effective that is considering that you can get better Water types to do that.
Without Huge Power, Azu moreso has to use its 100 HP to tank hits. After a very brief look at the Gym Leaders and Elite 4, I think Azu can be useful against Wulfric, especially if it has Sap Sipper for Abomasnow. Cryo can't do much damage to Azu with only Ice Beam and Flash Cannon. Depending on your options for Curse Avalugg, you can use Rain Dance Azu to clear Hail if Cryo set it up to deny Ice Body, then use Defense Curl, Charm, or Toxic to stall it out. It also gets Encore as a disrupting move that you can generally use. In the Elite 4, its Water type and decent bulk can help you against Malva, and it basically walls Diantha's Goodra.
These are the most notable things I've found with it, but I guess it all depends on what mons you have in the back. However, I think Soloing Wulfric's Avalugg is its best niche if you don't have a strong special attacker with Super Effective moves.
(Also I have no nuzlocking experience so this may just be dogcrap but lets see)
