The Runner Duck I posted earlier was too hard for the 2 year old grandniece to operate. It is tall and narrow and she kept tipping it sideways.
This is a low C.G. hen made for her. It is based on this image posted on line:
The outline is cut from poplar 5" tall, then 2 copies of the body below the neck are cut and glued on either side. That gives is some width so it doesn't tip too easily. Then 2.5" wheels, with the center holes offset 1/8" for a 1/4" dowel. The offset makes her waddle as she rolls. The push stick is a 3/8" dowel with a small wheel glued to the end (anti "poke-in-the-eye" tip). Spray painted Krylon Red paint+primer then trimmed with Krylon Canary Yellow for the wheels and beak. The eye is a 1/4" dowel rounded on both ends.