You're given an everstone for free by Ilima if you go to his house at some point in the game, I didn't know this was a thing until postgame but the unlock condition could be any point after completing his trial.
You can also buy Everstones for like 16 BP at the Battle Royal Dome. And they may be held by wild Pokemon as in the past, not sure.
Roggenrola hold Everstones at a 50% rate just as they have in the past.
They're found in abundance on Ten Carat Hill towards the end of Melemele Island.
Geodude have them as well albeit at a more typical 5% rate.
After going through the raw encounter table dump from @SciresM over the weekend it now turns out that Goodra, Garchomp and Vanilluxe are NOT exclusive pokemon as I had believed. And, as a result, presumably Flygon, Metagross and Gigalith are in the clear as well.
As it turns out, Goomy, Gabite, Vanillish and Castform show up in the wild as part of an additional SOS encounter dynamic I'm not entirely clear on.
As such, I would submit that the ONLY version exclusives are as follows...
SUN
Alolan Ninetales
Braviary
Buzzwole
Carracosta
Kartana
Passimian
Rampardos
Solgaleo
Turtonator
Whimsicott
Lycanroc (superior midday form) - can be caught in Moon, just not evolved there
MOON
Alolan Sandslash
Archeops
Bastiodon
Celesteela
Drampa
Lilligant
Lunala
Mandibuzz
Oranguru
Pheromosa
Lycanroc (inferior midnight form) - can be caught in Sun, just not evolved there