JENSEN BEACH, Fla.—Omar Mateen visited a gun shop in this town near his home about five or six weeks ago. His request: heavy-duty body armor—rated IIIA, the sort typically used by law enforcement—and bulk ammunition.

While at the store, Mateen had a phone conversation in which he appeared to be speaking in a Middle Eastern language, according to Robbie Abell, the co-owner of the store, Lotus Gunworks of South Florida. Staffers also noticed him texting on his phone.

Mateen “just seemed very odd,” Mr. Abell said. “The questions he was asking [about body armor] were not the normal questions a normal person would be asking.”

Lotus Gunworks doesn’t carry armor. Employees declined to sell Mateen bulk ammunition, turning him away. “If something is suspicious, it’s our discretion,” Mr. Abell said. “We are the gatekeeper.”

The FBI is in the process of reconstructing all of Mateen’s movements ahead of the shooting at an Orlando gay nightclub that left 49 dead and 53 injured early Sunday. The FBI is also looking into other parts of the gunman’s life, including his time at a corrections-officer academy. In 2007, he was kicked out of the school after an incident that led authorities to escort him off the campus, according to three former trainees in the program then.