Final answer:
The homograph that fits both blanks is 'record,' serving as a verb meaning 'to capture data' and as a noun referring to a vinyl disc for music playback.
Step-by-step explanation:
The homograph that can fill both blanks in the sentence 'The clerk had to _____ the sale of the new _____ I bought at the music store' is 'record'.
Here's how it works in context:
The clerk had to record (verb: to write down or capture data) the sale of the new item.
The new item I bought at the music store was a record (noun: a vinyl disc for audio playback).
The word 'record' is a homograph because it has the same spelling but different meanings and pronunciation depending on the context.