Are you kidding me? I already have trouble collecting exotics and (to be fair I also really want the last word anyway) and now I need it just to get a quest for a shotgun? Are you sure there isn't more to it, or is there just some glitch going on that's making it to where people can't get it?

Sorry to report I'm not. I just got the quest last night after finally finishing the main story (been working on mastering the third subclasses), so I can confirm she does indeed require the Last Word. I'm not sure how many stages there are to the quest (I'm assuming more than one, because that's par for the course with TTK so far), but you need the Last Word right off the get for Jolly Holliday.

So if I didn't have the last word before I completed the story, then I can never get the quest?

I'm not entirely certain, because I'm not sure if Jolly Holliday triggers on the character having Last Word (I'm assuming it does - makes sense). I think it may trigger once you do get the gun regardless of where you are in the TTK story. It's not a discoverable gun, like Thorn or Bad Juju or Invective (all of those that are tied to weapons bounties) and Xur has sold it in the past. It might work out that he either sells it again or, if you're able to get the Three of Coins from him before Bungie nerfs that buff, that you get a drop out in the wild off an Ultra.

Well funny thing about that one time xur was selling the last word a few months ago, I could have been able to have gotten it with the strange coins I had at the time, but unfortunately I was at a convention for the weekend he was selling it