Zyle is an NPC on the path through Gerudo Canyon towards the Gerudo Canyon Stable. He is near the cooking pot. If you talk to him, he will tell you that if you can bring a horse to him, he will buy it.
The nearest stable is the Gerudo Canyon Stable, but there's no wild horses around there, so if you want to go find a wild horse and bring it to him, you will have a long ride that you have to keep your horse safe through.
Alternatively, you can take one of your registered horses that you are no longer want to keep and ride it from Gerudo Canyon Stable to where Zyle is standing. The final method is to register a random wild horse at any other stable, then fast travel to the Gerudo Canyon Stable and ride from there.
Note: If you choose to take one of your own registered horses from the Gerudo Canyon Stable to save on time, make sure you pick one you won't miss because you cannot get that horse back again.
If you do choose to go from Gerudo Canyon Stable, there will be a wall of rocks your horse is unable to pass. Use Magnesis and Stasis to slowly clear the way and then your horse can pass through.
Talk to Zyle and sell the horse for the promised gold Rupee worth 300 Rupees. Zyle will then take his leave and this sidequest will be complete.