Amazon.com’s recipe for the department store of the future includes algorithmic recommendations and what one corporate director called “a magic closet” in the fitting room.

The online retailer is making another push to grow its fashion business, announcing on Thursday it will open its first-ever apparel store this year, with a tech twist.

“We wouldn’t do anything in physical retail unless we felt we could significantly improve the customer experience,” said Simoina Vasen, a MD...