For me it’s always been when they’re at my campsite and I go talk to them. If I talk to them after they have nothing to say or nothing I need to give them, they’ll ask me to make the item. I am unsure whether or not you can get them to ask you for it if they are not at your campsite.
Once they reach 15, you have to bring them back to your campsite if they're not there already and talk to them to bring their special item in the catalog so you can craft it. Annoying, I know. While it's crafting, you can kick them out and bring it back when it's done. Also, the item has to be out in your camp when you talk to them for the request to finish.
Note that it's highly recommend that their respective tier 3 amenity is unlocked so their points get counted.