Initiate and Recruit's backpack.
You… sort of can.
Achievements and collections are account-based, not character-based. So you can progress any achievement on any character.
However, in the case of the collection specifically tied to the order back items, they require a number of “items” which are gotten from completing story missions for HoT, all the way up through the final mission. And which items you get correspond directly to the back item you initially chose in Prisoners of the Dragon. So for example, a character who chose the Initiates back item will get collection pieces that correspond to that order’s exotic-level back item.
So you’ll pretty much need to finish the story on both characters if you want both back items. There are some other items that you need to buy from various HoT vendors, and those can be purchased with any character.
Once you get past the second / exotic level, then you get a second collection for the final, ascended back item. That one doesn’t require completing story missions or anything, so that one can basically be done on any character with the exception that each one requires you to also complete the collection for the respective Order’s armor and weapons, and you need to be a part of that order to purchase weapons and armor for that Order.