Not audio files, but Destinypedia does have text transcripts:
http://destiny.wikia.com/wiki/Story_Mission

There's a blue bar at the bottom of the page that says "Story Missions" - you have to click the [Show] link on the right side of that bar and it will give you links to all of the missions.

I don't know why that box is underneath References and isn't expanded by default…

Edit: Looks like several transcripts are missing, including everything from The Dark Below. :-\

I can't seem to find a place where all the Destiny dialog has been captured. No, I don't mean what the merchants say when you're walking by, I mean everything your ghost says, that you say, what those that send you on missions say. Anyone know if this has been captured anywhere? I've searched in the past, and again today, and all I can find are articles on Peter Dinklage :-\

You know, I COULD do this for DBO since I can capture and isolate the center channel. Like the Dialogue databank on HBO, except clean since only the voice is on the center.

Is there interest?