Actually, such a program exists for human characters: Makehuman. It is open source and there is a pipeline to import the characters via Blender into UE4. Animations have to be created in Blender either by hand or via Makewalk (separate project by different author.)
While I don’t have the programming/artistic skills to do such a task, I am certain that the code could be adapted to generate quadrupeds. The hard part would be creating all the mesh targets that would be necessary.