After spending 3 hours deciphering how this mechanic works, I figured I would write up a quick guide on how to catch the weather sos exclusive Pokémon.

You'll want adrenaline orbs which are purchasable from the vendor in pokecenters, a Pokémon with Rain Dance, Sandstorm, or Hail (depending on what kind of Pokémon you want), and a Pokémon that can stall with buff/debuff moves and can easily oneshot whatever the wild Pokémon calls to help.

The Pokémon which are exclusive to weather SOS are: Castform (Rain, Hail), Goomy (Rain), Vanillite (Hail), Poliwrath/Politoed (Rain), and Gabite (Sandstorm).

First, you'll need to weaken the Pokémon to 50% or less HP. Make sure not to inflict a status like sleep or paralysis, as this stops the Pokémon from calling for help. If you're catching a hail/sandstorm Pokémon, pick an opponent that won't be killed by the weather you're using.

Next, start the weather you need. Give your Pokémon the rock item which will extend the relevant weather, available in the room next to the Malie City flea market. Finally, use an Adrenaline Orb. This will increase the wild pokemon's propensity to call for help.

Now, get comfortable. Switch to your stall/oneshot Pokémon, and wait for the wild mon to call for help. Keep knocking out whatever it calls that's not the desired rare spawn. Refresh the weather as soon as it ends, as the special Pokémon only come during their desired weather.

The chance to encounter one of the weather exclusive Pokémon seems extremely low, but don't get discouraged, this DOES work. I got Castform from a Pelipper battle in the patch of grass outside Tapu Village, and Vanillish during a Sandshrew battle in the grass right before the Pokémon League on Mount Lanakila. Make sure to look up the area your desired Pokémon is available before starting a battle.

Good luck, and happy hunting!!!

Just wanted to add that Absol is a really good pokemon for this because it has access to all the weather moves and false swipe so load up on the weather move you want and false swipe and two attacking moves you got access to and go to town.

Another good pokemon to use is a toucannon when trying to get gabite because dugtrio knows only ground attacks and sucker punch and as long as you keep spamming the adrealine orb you wont get hit by anything exept the sandstorm and then you just have to refresh every 8 turns with absol you wont even need to kill any of them they will do it for you with bulldoze and magnitude.

oh and you can also get poliwhirl in the garden too aswell as a vanilish on mount lanakila and a sliggoo on exeggutor island

@coolbond: Throw on some Safety Goggles and sandstorm is a non-issue

@quid_pro_bono: I wasn't aware of these so thanks for introducing them. There are a few more you don't list here according to serebii. http://www.serebii.net/sunmoon/sosbattle.shtml

They don't list which weather effects you need. How did you find out?

@spyrodabs710: @coolbond: Thanks for the tips! I'm going to go try what you suggested.

@alistercat: It's pretty intuitive based on their typing. Poliwhirl, Politoed (night only), and Poliwrath (day only) show up when it's raining. Gabite appears during sandstorms (possibly only when the caller is Trapinch), and Goomy/Sliggoo during rain. Vanillite/Vanillish during hail. Castform in either rain or hail.

Hope that clears things up.

@milijango: Thanks. I dont know some of their types so I couldn't assume.