Setup:
I've run into a situation wherein the game I'm running the party wants to go to a nearby city and report on a school of Sahuagin kidnapping people along the coast and forcing them as gladiators in their area.   The party just escaped from the Sahuagin undersea lair but was not strong enough to take on them on and feels this is something the Lords' Alliance should be notified about.  (At least of all the factions in the Faerun, the LA seems like the most likely to report this too. At least I think so.) 

For those not familiar with Lords' Alliance

The Lords’ Alliance is an association of rulers from cities and towns across Faerûn (primarily in the North), who believe that solidarity is needed to keep evil at bay. The rulers of Waterdeep, Silverymoon, Neverwinter, and other free cities dominate the coalition, and all lords in the Alliance work primarily for the fate and fortune of their individual settlements.

Alliance agents include sophisticated bards, zealous paladins, talented mages, and grizzled warriors. They are chosen primarily for their loyalty and are experts in observation, stealth, innuendo, and combat. Backed by the wealthy and the privileged, they carry fine equipment (often disguised to appear common), including large numbers of scrolls scribed with spells of communication.

Agents of the Lords’ Alliance ensure the safety and prosperity of civilized Faerûn by standing united against the forces that threaten civilization. They proactively eliminate such threats by any means, fighting with pride for the glory and security of their people, and for the lords who rule over them. However, Alliance operatives are often glory hounds, looking to gain a leg up on their counterparts from other Alliance cities. The leaders of the Alliance know that the order will survive only if its members support each other, requiring a balance between pride and diplomacy. Rogue agents within the Lords’ Alliance are rare, but defections have been known to occur.

Question:

More of looking for a second opinion or a correction to my assumption.

Based on what I've read in the Basic Rules and Sword Coast guide would the Lords' Alliance have branches in various cities throughout Faerun or would it be more likely in the only 'membership' cities like Waterdeep, Neverwinter, etc?  I did not think it would be city watch as they oversee the law and order of the said city.  If not the Lords' Alliance who would the party report this type of 'crime' to?  

couple things (all imo)

• Think of the Lords' Alliance like NATO. Its designed to support member states.
• If it didn't happen IN an area governed by a member of the Alliance, I don't think they'd do more than keep an eye on it....even then, if it's a truly local threat, I still don't think they'd do more than observe.  I'd focus on the 'that threaten civilization' in your quote. IRL, you don't see NATO responding to the active sex slave industry in the US.
• For factions, I think the Order of the Gauntlet would care the most. They're all about rooting at out evil and wrong-doers...the whole 'zeal and justice' part of their description. (also imo, that's largely why WotC created the Order for 5e, as the whole 'i just want to fight evil no matter where its at or where it takes me' factor wasn't really covered by the existing factions)
• For other entities, whatever local authority governs the land the incident is happening in.  That could be 'none' though.

Most factions have agents, point of contacts outside their respective seats of power.   Realistically, the PCs would need an established relationship with LA to know the protocols of looking such a contact up in a non LA city.  If someone has the Faction Agent background and chose LA, they're set.  

The way to think of the factions outside of their respective domains is like intelligence agencies (the Lords Alliance is more like the UN Security Council, which relies on members states good will sharing of intelligence, but the analogy still works).  There's probably one or at most handful of folks who are like "real" LA in the city operating a "station" they maintain a safe house, and probably another location for them to do their business.*  They then cultivate people in the community they're embedded in to keep tabs on what's going on where they're at, and maybe even some assets who would go further and actual commit actions on behalf of the LA.  They're not going to have an embassy or an office that says "Lords Alliance" on it, unless the local power has welcomed the LA in as a sort of ambassador (LA, and maybe Zhentarim are really the only factions I can think of doing that, the rest are explicitly secret societies, the LA represents established powers). 

If the characters don't have contacts, it's probably best to report this info to local authorities, maybe local mariners guild (since it's safety info), sympathetic temples, etc.  If this is the sort of thing the LA may be interested in, if they don't know about the Sauhaugin already, maybe an agent will request to debrief them either through official channels (summoned to the city authorities to meet) or covertly (approached in a tavern by someone who wants to talk).  

*if you're familiar with Star Wars: The Force Awakens.  Think of Lor San Teca, the guy with the map chip, and his relationship with the Resistance.  Poe is a member of the Resistance, his contact actually found the map/information and is passing it on to the actual resistance agent.  Same goes for the respective First Order and Resistance agent's who reported in that Rey, Finn, and BB-8 were at Maz's castle.  They didn't do anything other than pass on word, both factions took such word as cause for military action.  If you want a hard core deep dive into IRL intelligence tradecraft, The Good Spy is a very accessible book on a real life CIA agent to eventual station chief's career in the Middle East (also really valuable insight into a character who may be a 'state actor' on behalf of the U.S./CIA, but has sentiments that actually go against official U.S. policy - a lot of field actual CIA agents in the 60s-80s were very much pro Palestinian cause despite official U.S. policy in the Middle East which largely disregarded them during the Cold War.  I've used that sensibility to add an element of "real spy stuff" to Burney in BG:DiA, and she's evolved to be the most important patron in my game, though Bahamut would be pissed if he ever found out the full extent of what she's been up to.)

The Harpers could also be interested depending on who the kidnapped are. If the victims are peasants, citizens, and other common folk, the Harpers would very likely get involved. I imagine a Harper agent or two would get themselves "kidnapped" on purpose and spearhead a gladiator revolt from within.

Depending on what direction you want the campaign to head towards, you could get all five factions involved if you really want to. By adding some details and incentives, all five factions could have some stake in it. For the Emerald Enclave, they are pretty sympathetic to the weak civilized peoples and often help them survive the wilderness and against natural threats (the sea and Sahuagin in this case); if you want more of a reason for them to get involved, you can have the gladiator fight be a form of ritual sacrifice that can cause disasters that mess with nature. For the Zhentarin, long term profit is all they care about, and rescuing a noble that is held hostage and make them feel indebted to the Zhentarin could open up more lucrative business opportunities.

If I were an adventurer, I would get as many factions involved as soon as possible. The more support I get, the more likely I am going to succeed on my mission and not die.