Why does the sniper shoot the man in the armored car and the woman?

In the story, the sniper shoots these people because they are his enemies. Remember that "The Sniper" is set during the Irish Civil War, when the Ireland was divided into two factions. The sniper is a Republican, while those inside the car and the woman below are Free Staters. The conflict between these two factions not only provides the context for the story, it also drives its plot.
Additionally, the sniper wants to shoot at the armored car because he knows that the people inside are looking for him. As Free Staters, they want to protect their own snipers, and they will do this by killing the Republican sniper.
Similarly, the old woman is also a Free Stater. In fact, she is an informant, and she tells the people inside the armored car about the Republican sniper who is hiding on the rooftop.
The sniper shoots her because she is a Free Stater who gives away his position and, more importantly, because she wishes for his death.