As the Animate dead spell statuses
A skeleton can only be created from an almost intact corpse or skeleton. The corpse must have bones.
A zombie can only be created from an almost intact corpse.
Most traditional taxidermy methods involve stretching the skin extracted from the animal on a mannequin, clay, wire, wool or some other material. Although some methods incorporate some of the animal's bones (such as the skull) into the manikin, it is unlikely that a stuffed animal would count as an "almost intact corpse", and it certainly would not have a complete skeleton.
If your DM allows you to animate the skin (strictly a homemade fault), you should remove it from the inert mannequin in some way, but you will end up with a truly horrible undead subject.