You find them in places like the factory that you raid in the beginning or hidden around the town hidden in rocks or in plain sight as a ball. It is best to check every rock like object, or narrow dead end that you may see in an alleyway.
Below is a list of locations that contain ability capsules which I got from here:
-Ability Capsules
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Peridot Ward
Hidden in a barrel outside the Mosswater Factory in Peridot (unreachable after finishing the raid with Julia, Fern and Florinia!) Obsidia Ward
Speak to a woman in the Obsidia PokeCenter looking for her son, find her son outside the Glamazonia Salon in South Obsidia and go back to the woman to reciece an Ability Capsule (new E15 event) North Obsidia
take the path that goes right, at its end turn down and it's hidden on the rock before the crack Under the Grand Stairway
choose the right path in the ledges puzzle and follow it through stairs and turn untill you reach the capsule (visible overworld item) Citrine Mountain
enter from outside of Shade's gym and go straight to the right, it's hidden in the last rock in that direction Byxbysion Wasteland
after going through the building with the stairs don't slide any of the steel beams, jump from the little one-square ledge at the upper left side and find it hidden in a barrel
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Subrailnet
after the Yureyu building when you enter the right side of the Underground Railnet, solve the rails puzzle to open the left door and enter the Subrailnet and it's behind a Rock Smashable shadow on the wall Byxbysion Wasteland beyond the Tunnels and Grotto
hidden in a bush in the Secret Power / former Mudkip / now Squirtle location, check the TM guide for more info and a video guide, requires defeating Aya to use Strength and ver very difficult to reach without Flash
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Chrysolia Forest
hidden in the bush at the end of the corridor at the top left, outside of the Moss Rock section Mysidia Railcave
hidden in a rock in the Confide / Ill-fated Doll Location, go through all the doors that lie between a pair of rocks to reach it
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7th St. Ability Capsule Vendor at the top left of 7th Street Agate Circus Ability Capsule Vendor at the lowest row, opposite of the Rest Area entrance Pickup ability (lvls 61-80:rare, 81-100:uncommon)