We can make them now. They take some rather hard to obtain items, but with some time invested you should be able to obtain them without too much trouble. The only item that could cause problems is the Resolve's Bridle which is obtained from Red Death, and that's only if there's no one down in Bedlam to help with crowd control.
As far as what you can do with them, they work just like an Animal Trainer at the stables, allowing you to claim or stable pets from your home. They are rechargeable with a Hitching Rope. It takes a Tinker to make both the rope and the post.