From the Gear icon in the upper right, choose Profile > Go To Artifacts Showroom. Your equippable (Chaotic) artifacts will be shown there once you have some. Right now you don't own any.

There's some cheap 'starter' artifacts in the marketplace, as well as a more expensive one for Time On Site quests that can be upgraded multiple levels (see the FAQ for details).

There's a nice FAQ about artifacts by MeguminShiro here: https://www.alienwarearena.com/ucf/show/2167784

I'm missing something here. I go to the artifact showroom page. I see a line that says "No artifact equipped" three times. I click on the Chaotic word and it drops down a list of categories: Weapon, Clothing, Power, etc.

I click on the category I want to see and then I see the cards that I have discovered. I click on a card and it shows me the information about the card and when I earned it. From this point I don't know what to do to "equip" the card.

I do have 17 cards so far.

Only Chaotic artifacts can be equipped. The other cards you see are not Chaotic ones. They're from 'achievements' and can't be equipped as they have no affect on ARP. The last of my first 3 comments apply if you want some Chaotic ones.