I'm not sure what you mean by "kill." O'Brien doesn't use the active verb "kill." It's the passive verb phrase "was shot," "was dead," and "was shot and killed." All of it is in passive voice.
The first mention of Lavender's death is on page 2:
Ted Lavender, who was scared, carried tranquilizers until he was shot in the head outside the village of Than Khe in mid-April.
The death is described matter-of-factly, in passive voice. O'Brien describes the death as if it were an everyday occurrence, like the rain or a new mission. It's part of war. It's something else to carry.
There is no one physically responsible for the death in Alpha Company. We must assume that a VC sniper took Lavender out with a head shot. Morally, all the men feel guilty for it, Cross the most.
Later, the third-person omniscient narration says on page 7:
...now Ted Lavender was dead because he lover her so much and could not stop thinking about her.
So, it is Lt. Jimmy Cross who feels the most guilt for Lavender's death, not O'Brien. He carries the weight of the dead body literally and figuratively (on his conscience):
But Ted Lavender, who was scared, carried 34 rounds when he was shot and killed outside Than Khe, and he went down under an exceptional burden, more than 20 pounds of ammunition, plus the flak jacket and helmet and rations and water and toilet paper and tranquilizers and all the rest, plus the unweighted fear.
O'Brien has Lavender killed to serve several purposes: Lavender is a symbol of unweighed fear. The function of Lavender is to be a doppelganger, a ghostly twin to haunt Cross (a Christ-figure), the one who feels the most guilt for his death. The novel begins with a death here, much like Hamlet does with the Ghost, and Lavender's death hangs over the story and novel as a whole. Like the Ghost in Hamlet, Lavender will keep reappearing. A major motif, as you know, is the ghost in The Things They Carried. Observe some of the titles of the other stories: "Ghost Soldiers"; "Lives of the Dead"; "The Man I Killed."
The purpose of storytelling, according to O'Brien, is to bring the dead back to life through memory: it is to resurrect ghosts. That is both the painful and joyous task of a writer: to turn a war story into a love story by honoring the dead.