Ok so I am gonna make this quick. They don't have their quirks here but they do have something that replaces their quirks. You'll understand once you read the story.

Here is a little description of the characters rolls in the story so there is no confusion so yeah.

Izuku "Deku" Midoriya: A Kind traveller who lives with his mom in a small house in the forest

Ochako Uraraka: A young girl who loves science but sometimes goes along with Izuku on his travels. She also has a crush on him. 

Katsuki Bakugo: Also known as Kacchan, He is a hunter who has a very aggressive nature. He is almost always accompanied by his best friend Kirishima

Shoto Todoroki: A young Prince who likes to sneak out of his palace to have fun and be away from his strict father the king. He loves riding on his pet horse Lulu

Tenya Iida: A brave knight who is close frinds with Izuku and Ochako. He is also Prince Todoroki's attendant along with Yaoyorozu

Eijiro Kirishima: A hunter and Bakugo's best friend. He is calm most of the time and is always there to keep Bakugo out of trouble

Momo Yaoyorozu: Momo is a female knight who - along with Iida- keeps an eye on prince Todoroki. She is also very rich but no one knows where she really comes from

Tsuyu Asui: A young waitress who used to have a crush on Izuku but now tries her best o make Ochako confess her love. She has a very good memory

Kyouka Jirou: She is a busker who met Momo in a bar at one of her gigs. 

Denki Kaminari: A goldsmith who is in love with Kyouka Jirou. He is a very good salesman and sometimes tricks people into buying his fake jewellery

Mina Ashido: An armourer who likes to call herself "The Book of Love". She works with Sero and is childhood friends with Kirishima

Fumikage Tokoyami: He is a falconer who owns a giant bird named Dark Shadow. He is a good friend of Asui

Hanta Sero: A blacksmith who works with Mina. He has a whole collection of decorative tapes

Mashirao Ojiro: A messanger who somehow was born with a tail 

Yuga Aoyama: A popular tailor that everyone knows. 

Mezo Shoji: A fishmonger who is a good friend of Tokoyami and Izuku

Koji Koda: A veterinarian who takes care of Lulu, Todoroki's horse

Rikido Sato: He is a baker who is very popular for his cakes

Minoru Mineta: Mineta is a cobbler who women avoid for valid reasons

Toru Hagakure: A former thief who was said to be able to turn invisible

Ok I am gonna leave it at that. As for the other characters outside of 1-a, well your gonna have to find out about them as you go along.