This page of the guide to Final Fantasy 7 Remake has a walkthrough for the Chocobo Search side mission available in Chapter 14 - In Search of Hope.
You start the side quest by talking to Stablehand. Your job is to find 3 Chocobo. While searching, you can move around the map using a fast travel option and perform other side tasks. Once you have found all of them, you will go back to Stablehand and complete the mission.
The first Chocobo is waiting for you in Sector 6 of the Slum - Collapsed Expressway - Old Bypass.
To find it, keep following the road until you come across a dark tunnel that leads you to the Chocobo. When you try to feed it, four enemies will appear. Your job is to eliminate them, otherwise, you won't reach the animal.
Opponents don't inflict heavy damage but they have an awful lot of life. Test their resistance with strong attacks and use Limit Breaks. That should weaken them considerably and allow you to win the fight.
The second Chocobo is waiting for you in Sector 5 Slum Area - Sanctuary Way.
Finding this Chocobo is relatively simple. All you have to do is get to the church and listen to the animal's cry. This time, there won't be any fights. Just run to the Chocobo and feed it to calm it down.
The third Chocobo is waiting for you in the Sector 5 Slum Area - Nuts 'n' Bolts Hills.
The Chocobo is waiting for you near the cemetery. When you get close to the animal, you will get attacked by an opponent. Defeat him to save Chocobo and complete the mission.
While traveling to this location, you can complete further parts of the Missing Children quest and Corneo's Secret Stash.
Reward: Free Chocobo rides.