Here is another way to think about Newton's 3rd Law.

The sum of forces in a state of equilibrium are zero. Essentially if an object is not moving horizontally, there are either no horizontal forces, or the forces sum to zero. In this case, the wagon accelerates because a horizontal force was applied (the child).

The child applies a force on the wagon, and consequently, the wagon does pull back on the child.

Now let's consider the wheels. There is a now a horizontal force applied to the wheels. Consequently, the ground applies a negative horizontal force (Friction). The wagon will only move when the force applied by the child moves is greater than the friction force.

An easier example may be to imagine a child and the wagon on a smooth (frictionless) surface, such as a sheet of ice. If the child pushed on the wagon, the wagon would push back onto the child, and both would move apart.