I'm aware the Chavin culture was a thing but i'm a bit unclear if they were an urban society or not and am having a hard time findiing information about them.
For that matter, i'm not that informed on any of the cultures I mentioned (I'm more of a Mesoamerica guy), and while i'm decently sure that the Moche, Wari, and Tiwanku were urban, I don't know how urban, and i'm unsure if they were actual state soceities or not (I'm sort of just assumiing they are, given the complexity of sites like Huaca de la luna); and i've seen Caral compared to Göbekli Tepe or Pueblo bonito, so i'm not even certan if it was an urban site/city or not.
You are a Mesoamerican guy, so I hope you know the scope of the question you're asking. I'm also in the midst of quals, so this is going to be more of a "go find it yourself, myah!" sort of response. My apologies.
You will want to look at Krzysztof Makowski's 2008 article on Andean Urbanism, which is a book chapter in the 2008 Handbook of South American Archaeology edited by Helaine Silverman and Bill Isbell. You should be able to readily find it. Makowski basically lays out the problem of talking Andean urbanism, with good discussion of the movement that some people like John Rowe argued for in the past, that the Andes were inherently "anti-urban". Makowski and I would disagree with this - however the nature of this urbanism is unique to the Andes. Depending on how much of a Mesoamerica guy you are and if your Spanish is good, Makowski also put out a beautiful volume in Peru on this topic. Really good pictures from this.
I would also argue/caution that contra some of the other scholars who have worked in Chavin, the best way to interpret Chavin de Huantar at its apogee is as a proto-urban center. Perhaps a few thousand people were there, but this was an accumulation of several hundred years of occupation by this point, and it could probably not be called a city as such.
As for the Moche, Wari, and Tiwanaku, there are many reasons to believe that the Moche were a set of peer polities perhaps approaching statehood (though this is up for debate on how unitary that phenomenon was), that the Tiwanaku was some kind of proto-state, and that (especially as excavations finally heat up at the capital) Wari was more than a state, but an empire. Again, I would point you to relevant chapters in the Silverman and Isbell Handbook for a good start on some of the recent-ish literature on this. Moche archaeology is also turning into the Andes' "Mayanists" phenomenon - mochecologos are having their own conferences at this point.
There's a lot of research out there on just these topics - I wish I had more time to answer your question, but the resources are there for you to explore this on your own.
Oh, and Caral is definitely not a city. That is something that /u/CommodoreCoCo and I have discussed in prior posts here on the subreddit.
EDIT: Forgot to add, for the Chavin phenomenon check out Richard Burger's 1992 "Chavin and the Origins of Andean Civilization".