Mostly you buy them, but they have also been on the calendar, Steam events, Discord, daily raffles, and daily quests. You don't need to "discover" them, they're usually advertised on the front page, or through relay posts.

Chaotic artifacts are only available to those who challenge Cartire to Kal-if-fee and survive.

https://www.alienwarearena.com/ucf/show/2167784 has a list of every artifact drop in the past and how they were obtainable. No easter egg mechanic.

@Punished_Pyle Wait how many marriages does he get through in a year that this is a regular occurrence?! :D