You need four mementos to get the pets. Those can be found in the areas or bought from Lost & Found later. It doesn't matter. When you get the memento, just go to the people and talk to them about it. Since you also need the other mementos for a Vigil, just buy all of them (or find). They doesn't cost much. Easy as that.
Important: Buy all the mementos before the Ura invasion. If you don't have all the pets at this part, there is a chance of missing the achievement. So just make sure you always buy new mementos after every mission.