Shake Shack Inc. hardly seems like a small enterprise, with 7,600 employees, about $500 million in annual revenue and net income last year of $24 million. Even so, it plans to apply for a new government-guaranteed small-business loan.

The New York-based fast-food chain says it needs the help to get through the new coronavirus pandemic. Many of its roughly 140 company-owned U.S. stores are in high-traffic urban areas now largely shut down by the virus. Sales are down 70% on average, the company said, and it has furloughed...