Yes, it's correct. Awkward because of the length of the parenthetical statement, but correct.
However, it might work better to use dashes:
If someone comes in and says – “Look, I need a new part. It’s broken.
I’ve got someone here that’s waiting for repairs. They want that part
as quickly as possible” – product availability is important.
Punctuating a transcript involves trying to decode the intent of the speaker while being as objective as possible. Part of the difficulty in transcribing is that spoken English doesn't conform neatly to rules: we typically speak in sentence fragments, and ellipsis (leaving out words that are understood in context) is common.
For example, in the above transcript, I could assume a consequential "then" is implicit - "If someone comes in and says X, then product availability is important." If this, then that. However, the speaker might not have meant this at all. The last four words might instead reflect the speaker suddenly inserting their own editorial comment about the quoted statement, and the opening six words of the sentence have been left hanging, the thought unfinished.
Using the dashes is one way to parse the whole paragraph without editorially resolving what the speaker's intent might have been. The awkwardness remains, but the transcript is as objectively valid as possible.