In "Hunters in the Snow," could the story have happened in warmer weather?

The cold-weather setting of "Hunters in the Snow" is a factor in the irritability of the characters, but it doesn't become essential until Rub and Frank are driving Kenny to the hospital. Instead of staying on the road and keeping to their mission, they keep stopping to warm up, and so become absorbed in their own issues instead of remembering that Kenny is dying in the bed of the pickup truck:

The wind was blowing into their faces. The snow was a moving white wall in front of their lights; it swirled into the cab through the hole in the windshield and settled on them. Tub clapped his hands and shifted around to stay warm, but it didn't work.

"I'm going to have to stop," Frank said. "I can't feel my fingers."
(Wolff, "Hunters in the Snow," classicshorts.com)

At this point, Tub and Frank reveal their secrets, which are partially motivating their actions. This allows them to bond, becoming less concerned with Kenny's well-being and more with their own -- they even take his blankets because they decide that he isn't getting any use out of them. Without the cold weather, they might have kept driving and not taken a wrong turn; Kenny's fate rests entirely  on the cold weather, which keep Tub and Frank from focusing on the goal.

In "Hunters in the Snow," how does the weather influence the story's events?

The story takes place in the winter, and the cold weather is seen often in the dialogue and narration. The first scene, which sets the tone for Tub's mistreatment by Kenny, focuses on how he waits in the snow because Kenny didn't arrive on time, and from then on, the cold snow is a major symbol:

The snow was shaded and had a glaze on it. It held up Kenny and Frank but Tub kept falling through. As he kicked forward, the edge of the crust bruised his shins.
(Wolff, "Hunters in the Snow," classicshorts.com)

Later, when Tub and Frank are trying to get Kenny to the hospital, they keep stopping. They make excuses, but the real reason they keep stopping is because of the cold; the truck window is broken and they can't stay warm. Their small bonding over personal issues while Kenny slowly loses blood is accelerated by the cold, and in the end they take his blankets for themselves and continue driving, not caring if Kenny freezes.