I am confused by this part にかけてきた. Can someone explain to me?
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「曲をかける」 means "to play a song/tune on the stereo/CD player, etc.".
In this case, the music is being played on the broadcasting system in a school during lunch time.
The music "reaches" the students while eating; therefore, it is described as 「かけてくる」 from their perspective, which becomes 「かけてきた」 in the past tense.
「この曲を給食のときにかけてきた」 is a relative clause that modifies the 「放送委員」.
"The broadcast committee member(s) who played this song during the school lunch time."