Where to buy gear pieces for your horse
Sometimes, you might obtain gear pieces for your horse through certain quests or item chests. However, the most reliable way to get equipment is to look for Saddler vendors. These NPCs sell all sorts of bridles, horseshoes, caparisons, and saddles, though these items tend to be very expensive, too.
Other merchant types, such as Grooms, Tailors, and Traders might have mount-specific items for sale, but Saddlers have more wares in their stock.