Okay, let's just be frank: The gun is a phallic symbol. Glad we got that out of the way, because we're blushing like crazy over here.
Although the story never directly touches on sexuality, it hangs out in our peripheral vision. To put it more simply, the story is concerned with innocence and the growth of a young girl, and sexuality is an important part of any human's growth into adulthood.
The first real indication of this in when we see "the woman's heart, asleep in the child, [...] vaguely thrilled by a dream of love" (1.26). The gun frightens Sylvia at first, but she stops fearing it around the time that she starts to develop her little crush. Coincidence? Yeah, we think not.
Again it's not about actual sexual exchanges between the hunter and Sylvia, but the fact that the hunter represents the human world's first temptation of Sylvia. In the natural world where she now exists, there aren't many options for human interaction, much less romance—and because of this, we can see a correlation between nature and innocence in this story. So the hunter and the lure to betray nature that he represents can be understood as a temptation to leave innocence behind. That Sylvia ultimately chooses nature is fitting—she is only nine, after all.
The gun, then, represents maturity on two levels: sexual (again, it's a phallic symbol, yo) and technological. The hunter commands nature to his own ends, he kills it and collects it, whereas Sylvia and her grandmother's relationship to nature is much more deferential and harmonious. Their lives are intruded upon, then, both by this sexualized masculine presence and the urbanization it's associated with.
And here you thought a gun was just a tool.