瞳に涙をためながら、おずおずと、彼が香織さんに近づいてくる。
I'm curious, what is the usage of てくる in this sentence? Why not いく? Given the character in question is clearly "heading" for the 香織さん and it's not 香織さん coming toward him.
Edit: it's narration, and the narrator is observer
It's the subsidiary verb (-て)くる, and it's used to describe 彼 is physically coming closer to 香織さん.
This may seem redundant at first because 近づく ("to get closer") also has a similar meaning, but they play different roles. 近づく and 遠ざかる are about the distance from the target (marked with に/から), while (-て)くる and (-て)いく are about the distance from the observer (which may or may not be the same as the target). For example, you can say AさんはBさんに近づいていった, which means A got closer to B and away from the (implicit) observer.
In your case, there is a narrator that describes this scene as a third person, but 香織さん is working as a temporarily observer here. Or you may think there is an invisible narrator (or camera) near 香織さん. See also: What does ～てくる mean here?
(-て)いる has a totally different meaning. It's possible to use (-て)くる and (-て)いる together, like this:
彼は香織さんに近づいてきている。
He is (gradually) getting closer to Kaori.
