If you are playing Animal Crossing: New Leaf (Welcome amiibo) in Citra, then you can see there are some extra options that are not in the original version. One of them is “Link HHD data”. If you also play Animal Crossing: Happy Home Designer in Citra, then it is possible to link its save data.
Any options will work. If you choose “retail game”, then Isabelle will ask if you insert AC:HHD on the 3DS or not. There is no “No” option if you got that prompt, but it’s OK, even if you can’t emulate putting game cartridges in it, because she’ll just check if there is save data or not (AC:HHD should be played once on Citra to create its save file). If you choose “downloaded game”, she’ll just skip the question and check for HHD save data.
Then Lottie will tell you about your gifts, and then the game will froze. Don’t worry, the game just got reset by itself (Citra will always froze if it got shut down or reset), but your save data is safe, and now you have the gift, and access to new furniture that you can buy from the catalog!