This Destiny: The Taken King Exotic Quest Guide will show you how to find the Chaperone Exotic Shotgun.
It's obtained once you complete the main quest line in The Taken King and become Rank 3 in the Crucible. The quest chain will obviously allow you to incur points, so it could be worth it for you in the end in that respect.
The chain starts off with Shindig in the Crucible, which tasks the player with defeating Guardians in the Crucible with the Last Word Equipped. You don't have to use Last Word, just have it equipped.
Finish the quest, hand it in, and the next part will kick off. It's called Fiscal Persuasion and the player must purchase a "generous donation" for Tex Mechanica from Cryptarch. This sets you back 15,000 Glimmer.
Again, finish the quest, turn it in, and start the next quest Buckshot Bruiser. In this quest, you will need to use a shotgun to defeat Guardians in the Crucible along with high-level minions of The Darkness. The Universal Remote also counts, so you can throw on two shotguns and make things easier for you.
Finally, once you turn in this quest the Two for the Road quest kicks off. In this one, the player needs to use a shotgun to defeat the Shield Brothers on the Dreadnaught. You cannot do this on the normal version of the Strike, obviously, because this quest line give the player a specific version of the Strike.
Arekkz said he's not 100% sure whether you will need to kill both Shield Brothers with the shotgun or not, but just to be safe, do it anyway. Only the killshot with the shotgun counts, so you can use other weapons to get their health down. This version of the Strike also includes matchmaking, but if you take advantage of this, make sure to secure the killshot. It would be wise do this with friends just to make sure someone else doesn't get the final shot in instead.
At last, you can return to Amanda once this quest is finished; however, she is going to make you wait. Thus, yet ANOTHER quest pops up. It's called Bide Your Time, which means the player needs to wait around on Tex Mechanica's committee to complete its selection process.
How long do you have to wait? Until this week's reset.
So, if you picked the quest line up today, you will have to wait until next Tuesday's reset. This is because the reset has already kicked off for the week of September 22.
Once you have waited an entire week, go talk to Amanda and she will finally hand over the Chaperone.
Now, here's the million dollar question: should you go through all of this to get the weapon? That's for you to decide, but according to the video from Arekkz below, the Chaperone does contain a few nice perks such as Solar damage.
Other perks include:
The Roadborn - Precision kills briefly grant bonus handling, range and precision damage.
The Survivor - Three rapid precision kills while Born on the Road is active will return two rounds to the chamber.
The Chaperone - Fires a single precision slug. Grants Agility boost.
Infuse - Consume a more powerful Weapon to boost the Attack value.
There are selectable perks as well such as: Accurized Ballistics which provides more range and Impact with increased recoil; Linear Compensator which has more predictable recoil and provides slight boosts to range and Impact with more recoil; and Field Choke with more range and Impact with increased recoil.
Secondary selectable perks are: Field Scout which increases carried Ammo capacity; Single Point Sling which allows you to switch weapons faster and move quicker while aiming; and Snapshot which provides incredibly fast aiming.
The video below goes into more detail on the Chaperone.