I never caught a shiny in Moon. I found a shiny Swellow in UM in ultraspace and currently I'm working on getting a shiny charm to get more.
I haven't found anything in moon and since I switched over to UM, I probably wont find any more. I did find a Shiny Sigilyph in ultra space and I'm currently breeding to get a Shiny eevee.
Not counting GA giveaways on Tumblr, my first shiny in Moon was a Goodra in a Link Trade. In Ultra Moon my first shiny (again not counting GA giveaways on Tumblr) was a Bonsly during a SOS chain while looking for a Happiny.
It was a cottonee, i was doing some competitive battle training and all of a sudden a shiny popped up and i ALMOST clicked attack.
My first shiny in Moon (also my first ever shiny) was an eevee that I hatched. I evolved him into an umbreon and now he's a competitive tank.
My first shiny in Pokémon sun was an Araquanid, purple spider was lit. So far no shiny in UM but planning on a hunt after I finish the main story. (So far behind)
My first shiny that I caught in Sun version was, surprisingly, a Minior. I was so terrified that it was going to Self-Destruct in my face just to screw me over again (Vietnam flashback of a shiny Weezing using Explosion first turn), but it didn't. And I managed to catch it without incident. I was so ecstatic.
My first Pokémon without using the Ultra Wormhole was a Pelliper which actually made me happy because I still remember it to this day, but back when I was playing White I found a shiny Wingull and had no poke balls of any kind so this Pelliper helped make up for that lost Wingull.
I never catched a shiny Pokémon in sun/moon. ...Although I believe I once saw a shiny Pikipek. But I once got a shiny Salazzle in a wonder trade, so I think my first shiny in sun/moon was a Salazzle.
Two so far, a Carbink (a favorite Pokémon of mine too) in Moon and a Caterpie in Ultra Sun. Both during nuzlocke runs no less, very happy with the finds.