The difference between what Orwell is supposed to do and what he wishes to do is the primary way that Orwell communicates his vague uneasiness.
In Orwell's feeling of being "vaguely uneasy," there is a clear gulf between what exists internally and what is seen as external obligations. This is evident when Orwell talks about the rifle, itself. Orwell makes it clear that he did not really wish to use the rifle for killing the animal. Rather, he says that "I had merely sent for the rifle to defend myself if necessary." This shows the difference between what others want him to do and his own perceptions.
Orwell knows that the rifle symbolizes violent action. To a great extent, he feels uneasy at how the rifle placates the village mob, especially in comparison to how he himself perceives its use. The two expectations are different: one is in terms of aggressive violence, the other is in terms of the desire for self-defense. Orwell feels uneasy about the mob's expectation because he expects the rifle will be for self-defense use, which is why he adds the "if necessary" at the end of his comment: "I had merely sent for the rifle to defend myself if necessary."
His unease is displayed because there is a clear collision between his own intrinsic expectations and what is extrinsically expected. Orwell further communicates vague uneasiness through how he sees himself in the eyes of the mob: "It is always unnerving to have a crowd following you." Orwell feels uneasy because he is constantly being observed by the mob. This observation causes his actions have unexpected resonance with the mob, even if he does not wish them to.
In the final analysis, no matter what choice Orwell makes he is going to be unhappy. If he refuses to kill the elephant the mob will denigrate him further, and if he does kill the elephant, he is acting in accordance to their wishes, not his own. His uneasiness at being perceived by the eyes of the mob echoes the conflicting expectations.
Finally, Orwell's vague uneasiness is seen the moment when he sees the elephant itself. Orwell recognizes the elephant is not a force of malevolent destruction. Rather, Orwell acknowledges an almost pathetic state:
The elephant was standing eight yards from the road, his left side towards us. He took not the slightest notice of the crowd's approach. He was tearing up bunches of grass, beating them against his knees to clean them and stuffing them into his mouth.
The elephant does not take "the slightest notice of the crowd's approach." He simply eats, living his own life. Orwell's unease results from knowing that he is going to have to do something that will constitute a violation of his moral desires. Upon seeing the elephant, Orwell is vaguely uneasy because he is convinced that his official capacity belies his personal convictions.