Can Azir's ability swap positions with a unit at Vilemaw's Lair, given that units cannot move from the Lair to base?

No. When Azir targets a unit at Vilemaw's Lair, Azir moves to the Lair successfully, but the unit cannot move to base due to Vilemaw's Lair's restriction ('Units can't move from here to base'). The unit's move fails and it stays at Vilemaw's Lair. Azir ends up at the Lair with the unit instead of swapping positions.