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.
D&D 5e related rules
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.
What does citizen mean?
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.
What about earlier editions?
Let's say we find this unsatisfying, and wish to look to see if earlier editions could shed some light on this.
The 3.5e accessory book, City of Splendors: Waterdeep has the following relevant information about Waterdeep citizenry:
The City of Splendors is home to over one hundred thousand citizens, growing to five times that number in the summer months. Among the city’s vast numbers, a handful of individuals have achieved particular prominence.
(Pg 13, Who's Who in Watedeep)
This passage in particular lends significant support to our "plain English" argument of inhabitance based citizenship as opposed to something more formal.
Further support is given with this passage:
[...] Waterdeep’s last line of defense is the most important—its ever-resilient citizenry. On more than one occasion, the inhabitants of Waterdeep have rallied to defend their city, reinforcing the City Guard and City Watch. [...]
(Pg 19/20, City Defenses)
The second sentence of this excerpt explicitly links inhabiting the city with citizens of the city, not making a distinction between them.
In addition to this, we can use the 2e sourcebook Volo's Guide to Waterdeep for some more evidence:
As has been said before, the people of Waterdeep are Waterdeep. No guide to the city is complete without a mention of the more colorful and influential folk who live there and shape Waterdeep into what the city is.
With Elminsters help, weve included a best guesses list here of the probable classes, levels, and alignments of some of the Waterdhavians Volo mentions. Adventurers be warned: Much of what is said here may be wrong!
This list is alphabetical by first name (or only known alias) because so many citizens of Waterdeep lack surnames. Only ability scores of 16 or greater are listed, and the standard Realms character statistic abbreviations are used.
(Volo's Guide to Waterdeep > Appendix 1: Folk of Waterdeep > Pg 214)
The fact that many citizens only have an alias or first name, supports our interpretation above, as registered citizenship in a city the size of Waterdeep would require something akin to a last name (or a place identifier)
The edition neutral sourcebook Ed Greenwood Presents - Elminster's Forgotten Realms, has only one mention of what constitutes a citizen, and that reference is for the town of Secomber (which is notably not Waterdeep):
A citizen is one who owns land, pays rent, or has resided in Secomber or within view of its walls for at least a season.
(Ed Greenwood Presents - Elminster's Forgotten Realms > 2. Laws and Orders> Justice > Secomber's Code of Justice > Pg 55)
It might be reasonable to apply this standard to Waterdeep, in which case, jointly owning Trollskull Alley qualifies the party as citizens all.
But in other real-world city state societies you had to be born in the city to be a citizen...
We can disprove that being born in the city is a requirement for citizenship this pretty easily with the following passage from Volo's Guide to Waterdeep:
VOLOTHAMP GEDDARM (CG hm W5; INT 18, CON 17). Volothamp, or Volo, as he is more commonly called, is the author of this guide and of Volo's Guide to All Things Magical, a suppressed work. He was born in a bog somewhere in Faerun. [...]
(Volo's Guide to Waterdeep > Appendix 1: Folk of Waterdeep > Pg 232)
Here Volo is listed as being a citizen of Waterdeep, but was not born in Waterdeep. Thus citizenship must not be reliant on being born in the city.
Waterdeep: Dragon Heist has some additional circumstantial evidence:
[...] The colors of the Dock Ward are burgundy and orange, and its mascot is a swordfish that has always been depicted as green for reasons lost to time. The folk of the Dock Ward take competition seriously, and they frequently draft their champions from the rough-and-tumble sailors who come to the city. (Some say they draft pirates, but that is pure slander.) Frequent complaints arise that these women and men are more citizens of the sea than of the Dock Ward itself. But if they register with a magister and pay taxes, they are as welcome to compete as any long-term resident of Waterdeep.
(Waterdeep: Dragon Heist > Volo’s Waterdeep Enchiridion > Dock Ward)
The fact that there are questions on whether or not someone's citizenship would be dependent on where they spend the majority of the time (sailors and the sea) supports our original interpretation of the much looser definition of citizen ship based on residency.