Can I use Azir (green) ability when I'm on Vilesaw's Lair?

Yes, you can activate the ability of Azir, Ascendant while he is at Vilemaw's Lair, but the movement effect will not resolve as intended.

According to RiftJudge FAQ #8912 and FAQ #8, the interaction works as follows:

• Valid Activation: You can legally activate Azir's ability targeting a unit at another location.
• Movement Failure: Because Vilemaw's Lair has the restriction "Units can't move from here to base," any attempt to move a unit from the Lair to your base will fail.
• Result: Azir will successfully move to the target unit's location, but the target unit will be unable to move to Azir's original location (your base). The target unit will remain at Vilemaw's Lair, and both units will end up at the Lair together.

This is consistent with Rule 356.3.e.11, which states that instructions that can be partially followed are followed as much as possible and ignored otherwise. Since the movement of the unit is prevented by the Lair's restriction, that part of the effect does not occur.