Late Monday police said a warrant on a charge of first-degree murder had been issued for Aaron Christopher Kelly. 38 He was described in a statement as "armed and dangerous."
He is wanted in the death of Justin Krumbah, 38.
The shooting was reported at 11:03 a.m., and the first officer arrived one minute later, Richland Police Department Cmdr. Chris Lee said, NBC News reported.
Before opening fire with a handgun, Lee said, the suspect talked to Krumbah. It wasn't clear what the conversation was about or whether they knew each other, he said. A motive in the shooting also wasn't clear.
"We don't know what the cause was," Lee said. We don't know if it was specifically intentional or targeted toward any one person."
Krumbah was pronounced dead at the scene, interim police chief Brigit Clary said during a news conference.
The suspect fled the store, and police said he was believed to be driving a silver 2005 Honda Civic tagged in Washington.
In a statement, Fred Meyer said the company was "deeply saddened" by the shooting and the death, who has not been publicly identified. It offered its "thoughts, prayers and support to all affected individuals and families during this difficult time."
The statement added that the company is cooperating with law enforcement agencies. The store will remain closed while their investigation continues.
The second victim, who was not identified, was undergoing surgery on Monday afternoon and remained in critical condition, Lee said.
The market is a little more than two miles west of the Columbia River and 220 miles east of downtown Portland.