Walmart Inc.’s Jet.com unit will soon be making same-day grocery deliveries in New York City, upping the fight for customers in a highly competitive market.

Jet will open a fulfillment center in the Bronx this fall to bring same-day and next-day delivery to the city, the company said in an emailed statement. The center will stock Jet inventory, which includes groceries and everyday essentials, as well as other select merchandise.

Rival Target Corp. already offers same-day delivery for in-store…