SAN FRANCISCO — The retail landscape is littered with the casualties of changing consumer behavior. Shoppers are bargain hunting online, department stores are struggling, and once-mainstay brands are closing out permanently.

Then there is Stitch Fix, a mail-order clothing service that offers customers little choice in what garments they receive, and shies away from discounts for brand name dresses, pants and accessories.

Despite a business model that seems to defy conventional wisdom, Stitch Fix continues to grow.

For the fiscal year that ended last July, the company recorded sales of $730 million. It has been profitable since 2014 and has raised just $42 million from outside investors, a relatively modest sum for a high-flying Silicon Valley start-up.

And while Stitch Fix executives say they have no specific plans to go public, the company is well positioned to file for an initial public offering as soon as this year.

Should Stitch Fix go public, it would be the biggest retail offering since Etsy two years ago. Perhaps more important, it would be a Silicon Valley rarity: a profitable company that did not raise money at a sky-high valuation, and one that could potentially tap the public markets at a price many times greater than its current value.

Stitch Fix is not the first company to try this business model. Similar start-ups, from clothing rival Trunk Club to the cosmetics specialist Birchbox, have found a market mailing consumers a grab bag of items and offering free returns for anything unwanted.

But many such start-ups have had trouble keeping costs down, and customers around. Nordstrom, which bought Trunk Club in 2014 for a reported $350 million, wrote down nearly $200 million from the business’ value last year.

“Those businesses have mostly struggled to grow and remain profitable over a long period of time,” said Mark Cohen, a professor at Columbia Business School who previously served as chief executive of Sears Canada. “Will a loyalist want to receive this every month? Is anyone interested in consuming this much apparel and accessories?”

So far, Stitch Fix has found success where other online clothing start-ups have struggled. To the company’s founder, Katrina Lake, success comes down to delivering what consumers want: making it easier to shop.

“There’s been a lot of innovation around being the cheapest or fastest,” she said in an interview at one of the company’s warehouses south of San Francisco. In her view, what was important was helping customers find clothing they liked without taking lengthy shopping trips and returning dozens of items.

Stitch Fix was founded in 2011 and was initially run out of Ms. Lake’s apartment in Cambridge, Mass. At first the company catered only to women, but has since expanded to offer men’s clothing, plus sizes and maternity wear.

Each box contains a handful of selections from trendy brands like Citizens of Humanity, Scotch & Soda and Barbour. Up…