The Rage of the Emperor
I want to start by saying that I in no way think this will be a 'GOOD' build. I'm just curious to see if it even works.
So I'm needing to do a bit more testing to see if this is even valid, but I noticed that Sated Devourer's Phantom hit would proc on-hit effects while Azir was attacking with his sand soldiers. The trouble is Sated Devourer on Azir was, ... well, stupid. Jungle Azir in general was/is pretty dumb. ...
However, now that the Phantom hit has been transferred over to Guinsoo's Rageblade, this could potentially be something, .. probably not something big, but something none the less.
Guinsoo's Rageblade gives a good amount of AP and attack speed, which are great on Azir. It gives a decent amount of AD, which is not so useful, but can still be good for those situations where you need to auto attack while your abilities are on cooldown.
As I mentioned, I haven't really done any actual testing with on-hit abilities in game yet. I'm honestly not too sure which items would even be worth it really.
My first thought was
&
Because it would be easy to be able to tell if they were actually being proc'd, but unfortunately they are melee only and Azir is a ranged attacking champion.
(However,
does not proc while Azir is attacking with his sand soldiers. It could be that he is considered to be a melee attacker at this point, so it might work, which would be AWESOME!)
Otherwise there are of course the following items.
>
,
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,
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,
,
, 
Should also proc the Furor movement speed on your choice of boots.
I'll try and test this out soon. If anyone else would like to try it out and wants to reply with your findings before I get around to it, that would be great.