I'm going to post a contrary opinion here. I'm going through this right now with my table, and what hasn't been mentioned here is this bit (emphasis mine):
Characters who place the security of the city and the realm ahead of
their own interests are invited to join this faction. Potential
recruits must be residents of Waterdeep.
The second sentence has been addressed well here, but I think not the first. This wording strongly implies to me that only lawful-oriented characters will get invites, as that's essentially a paraphrasing of the definition of lawful alignment.
This is further backed up by the fact that this is the only faction that explicitly requires accepting and completing its missions to remain in good standing. You gotta follow orders, or you're out.
Now, that's just to get invited. There are actually two described ways to join factions according to this module: to get invited after Chapter 2 begins (which is what this question is about), or to take the faction agent background at character creation, and pick that faction (as per the Factions in Waterdeep section of the Introduction).
I could see someone chaotic (or non-resident) picking that faction at character creation, with some interesting backstory as to how they ended up there. On the outside, I could see a (law/chaos) Neutral character who RPs closer to the lawful side getting an invite in Chapter 2 as well. But it doesn't seem like a Chaotic-aligned character could reasonably get an invite for this faction, as they don't in fact place the security of The System ahead of their other interests.