If I have Jax unmatched in base, and use his effect to play an equipment as a quick draw reaction, can I trigger Azir's legend ability to play a sand soldier token?

No, you cannot trigger the activated ability of the Emperor of the Sands legend in the same chain as playing an Equipment via Jax, Unmatched's Quick-Draw effect.

Reasoning

1. Ability Type: The ability on Emperor of the Sands ("[1], [T]: Play a 2 [S] Sand Soldier unit token to your base") is an activated ability.
2. Timing Restriction: Activated abilities that do not have the [Action] or [Reaction] keyword—like this legend ability—can only be activated during your Main Phase when the game state is "Open" (meaning the chain is empty, per Rule 150.2/148.2 and reinforced by the RiftJudge FAQ #9120 regarding Jax's legend ability).
3. Chain State: When you play an Equipment as a [Reaction] using Quick-Draw (granted by Jax, Unmatched), you are acting in response to something or initiating a chain, making the game state "Closed." Even if you were in your own Main Phase, the act of placing the Equipment on the chain (via the Quick-Draw trigger) creates an active chain, preventing you from activating a non-speed-keyword ability until that chain resolves and the state returns to Open.

In summary: Because the legend ability lacks the [Action] or [Reaction] keyword, it cannot be used during a chain, even if you have met the requirement of "playing an equipment this turn." You must wait until the chain has fully resolved and the game state is Open again to activate the legend ability.