The film's official novelisation explains this. After he arrived in the past, he had a few hours to kill while he was waiting to acquire Sarah. During that time he helped himself to a car, then broke into a sporting goods store to get hold of some ammo for the .38 handgun he took from the policeman and the shotgun he purloined from the cruiser.
Reese sank into the seat and leaned his head back. He felt his
muscles, knotted by tension and use, begging to relax. A wave rolled
up to the shore of his conscious mind and offered to carry him to a
warm, peaceful place for a few hours’ sleep.
No way. He sat up and emptied his pockets; three boxes of Super .38
ammo and four more of .00 buckshot. That would hold him for the time
being. He had broken into a sporting-goods store at 9:15, and by 9:16
his pockets were full, and the guard dog there was sorry to see him
go.
Terminator: Official Novelisation
Notably, he also got hold of a whole bunch more .38 ammo while Sarah was sleeping after their escape from the Terminator.
“What have we got here? Corn syrup, ammonia, mothballs. Umm. What’s for dinner?” Reese didn’t rise to the limp attempt at humor. He was unpacking another sack filled with ammunition for the .38, road flares, tape, scissors, a small pan with a strainer, and matches. “Plastique,” he answered, distracted.