I have an old 3DS XL, and I recently downloaded the new AC:NL update. I understand that the amiibo cards can now do some neat things, but I don't want to buy a $20 peripheral for 1 game. Can I just use a camera to scan the cards with the camera or do I need the reader/writer?
Amiibo Cards work the same was as regular Amiibo figurines, in that they require a special reader/writer. Meaning, if you don't have the new Nintendo 3DS that comes with this reader/writer built-in, you will need to buy the NFC adapter that you've mentioned.
For reference, see this tutorial on how to use the Animal Crossing Amiibo cards.
Namely, this step:
Follow the instructions and when prompted, place the Animal Crossing amiibo card of the animal you want to visit on the NFC area of the touch screen (New Nintendo 3DS system) or on the NFC Reader/Writer.
These are the only two ways to access the Amiibo Card, or an Amiibo, with a Nintendo 3DS for that matter.
It's worth noting that this isn't the only game that you would be able to use the NFC for. Two enjoyable games that you could use the NFC with applicable Amiibo for are Super Smash Bros. for the 3DS and Fire Emblem Fates.