With green Azir in Vilemaw's, if you do the action will it keep Azir in Vilemaw's and still move a unit into Vilemaw's?

Yes, if you activate the ability of Azir, Ascendant while he is at Vilemaw's Lair targeting another unit you control, Azir will remain at Vilemaw's Lair, and the target unit will also end up at Vilemaw's Lair.

According to RiftJudge FAQ #8 and FAQ #9347, this interaction works as follows:

1. Valid Activation: You can legally activate Azir's ability targeting a unit at another location.
2. 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.
3. Result: Azir will successfully move to the target unit's location (or stay at the Lair if the target is already there), but the target unit will be unable to move to Azir's original location (your base). The target unit's move fails, and it remains at Vilemaw's Lair. Both units 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.