After taming the horse and adding a saddle to it, you can add a chest by left or right clicking the horse, I forget which.
Upon death, the horse should drop the chest and the contents of the chest.
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First off, you can not put chests in horses. You will need to find a donkey varient. Donkeys are recognizable with their long ears. Then click the donkey with a chest.
You cannot place a chest on a horse.You can however, place one in the inventory of a Donkey.Simply sit on the tamed donkey and press E.This will open the donkey's inventory.Place your chest in the box that has the silhouette of a chest.
Open the chest and then you will be able to drag items into the chest, in which they will then be removed from your inventory.
press on the horse with a right click then you can put on armor and saddle on a horse
Hold shift and then right-click with the lever to place it without interacting with the chest.