They are only owned by the rich, because no one else can afford to use magic in such a trivial way. Or more specifically, the children of the rich.
They are usually sold with a velveteen-slave, whose has the job of ensuring the integrity and cleanliness of the stuffed animal.
As you would expect, they are fabulously expensive. Steal a velveteen from a princess and you're set for life. Many have killed and died to get their hands on a velveteen. It is difficult to sell one, however, since they are each unique and well-known (at least among nobles, who often show their wealth off to each other).
The details of their manufacture are a closely-guarded secret, but they are produced by the Armenjero "Empire", a cliffside mecca for bards and gypsies. It is famous for it's gambling, and it was won from an archmage of Meltheria in a game of chance.
One big caveat is that velveteens are condemned by the Church, which claims that they are animated by the souls of the dead. Velveteens are sufficient to warrant a full investigation by the Church's witch hunters, who will track down the velveteen, interrogate it (fruitlessly, because all they do is wiggle and play), and torture it before burning it to death.
The proof they offer is this: velveteens can be turned by clerics, just as undead can.
How To Use Them In Your Game
Add them to your loot tables. They're a hell of a lot more interesting than another emerald the size of a baby's fist, and worth about the same.
If nothing else, they can be used to set off traps or something.