amiibo cards came out with amiibo festival as a way of bringing ur favorite villagers into the game. they came in packs of i think 4-5, with 4 series. this
official site (click the words its a link) shows all the villagers per series
a few years into animal crossing new leaf, there was an update to the game that allowed u to use amiibo cards to "spawn" villagers from the amiibo cards into ur game, and thru that u could invite them to ur town. this update also released another series of amiibo cards called welcome amiibo that brought back 50 villagers from older games that didnt make the cut into new leaf. along with new villagers, this update also brought new furniture.
in new horizons, the amiibo cards work much the same. u use them to spawn in a villager. they'll only work after uve built ur campsite and invited the first villager that shows up to ur island. after that, spawning in a villager with amiibo cards will place them in the campsite. theyll ask you to do some tasks, and i believe ppl said u have to spawn them into ur campsite 3 times and complete their tasks 3 times before theyll move into ur town.
to get kody u have to have his amiibo card (or u could get him from someone else's town). if u dont have one, i imagine that the amiibo cards will be reissued for release soon since they function in new horizons. he shows up in the series 3 packs, so u could exclusively buy those for him. i also have him in my town, and if he asks to move id be willing to let u snatch him from me.