Do lux and Vayne live in the same place?

They both live in the city-state of Demacia.

Previously Demacia has been referred to as a single city, but with the Lore rework we've seen the city-state of Noxus become the "Empire Of Noxus", with the Noxus we're familiar with as the capital.

I wouldn't be surprised if they decided to do something similar with Demacia, but for now, we know that yes, they live (or at least lived, Vayne's sort of a wanderer) in the same place.

Demacia is probably still the capital and where most important people live, i don't think there are any other big cities in the kingdom.