Can I use Azir Descendant ability in Base to move a unit from Vilemaw Lair?

No, you cannot use Azir, Ascendant's ability to move a unit out of Vilemaw's Lair to your base.

According to the official FAQ regarding Vilemaw's Lair:

• Vilemaw's Lair restriction: "Units can't move from here to base."
• Interaction: When Azir targets a unit at Vilemaw's Lair, Azir will successfully move to the Lair, but the unit's attempt to move to Azir's original location (your base) will fail because of the Lair's restriction. The unit will remain at Vilemaw's Lair, and Azir will end up at the Lair with that unit instead of swapping positions.

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. In this case, the movement of the unit is prevented by the Lair's restriction, so that part of the effect does not occur.