If they weren't already citizens, owning property in Waterdeep certainly qualifies them as citizens, and thus qualifies them for membership in the Lords' Alliance.
How one becomes a citizen of Waterdeep is not defined in the game, and in particular it is not defined in these books:
- Sword Coast Adventurer's Guide
- Waterdeep: Dragon Heist
- Waterdeep: Dungeon of the Mad Mage
Given that citizen isn't a game defined term ,we need to rely on the English meaning of the word.
Merriam-Webster defines it as:
1: an inhabitant of a city or town
especially : one entitled to the rights and privileges of a freeman
Collins Dictionary defines it as:
2. countable noun
The citizens of a town or city are the people who live there.
The Oxford English Dictionary defines it as:
1.1 An inhabitant of a particular town or city.
All three of these dictionaries have citizen defined as someone who is an inhabitant of a particular town or city.
So for someone to be a citizen of Waterdeep, they just have to live there. If they didn't qualify before as residents of the city, once they own property and live in it they certainly qualify as residents of the city.