In a sprawling food processing plant on the outskirts of Nashville, Jon Matthews is expected to be everywhere. He oversees one million pounds of produce that flow into the plant daily to be sliced and separated into salads and sandwiches.

For months, Mr. Matthews’s employer, produce giant Taylor Farms, has been engaged in an all-hands effort to cajole its 22,000 employees to be vaccinated against Covid-19, offering on-site vaccination clinics and cash incentives. The company founder, Bruce Taylor, recorded a public-service announcement that plays in the hallways, employees said.

Mr. Matthews, 50, is among the many unpersuaded. The inventory supervisor is uncomfortable with the novel technology behind the vaccines and remains unconvinced it will protect him from infection, particularly in light of the Delta variant, which appears to break through faster than earlier strains.

“Thirty percent of me says it might be the smart thing to do,” Mr. Matthews said.

Mr. Taylor, his boss, has no such doubts. “I believe the vaccine is a miracle,” he said. “Why wouldn’t we take advantage of it?”