If they were both floating in space, you'd have a situation similar to what you're thinking. Both the wagon and the child would accelerate towards each other. However, the wagon would move faster, because it (presumably) has less mass than the child, so the force accelerating the wagon would have more effect than the equal and opposite force accelerating the child.

They are not both floating in space. The child's feet are on the ground, and the wagon's wheels are on the ground. They are both accelerating towards the Earth's center of gravity. This means while the child experiences a horizontal force, that force also acts on their feet which are acting on the ground. Friction between the child's feet and the ground causes the child to stay in place. On the other side, the wagon's wheels do not produce as much friction and roll freely, so it moves. (Technically we could say the forces here have an impact on the Earth's motion, but in this case the difference in masses is so great we can ignore it.)

Imagine if you take the wheels off of the wagon and load it up with 400 pounds of concrete. Now something very different happens: the child pulls, but the wagon stays still. In fact, the child will very strongly feel the friction of their feet on the ground, and if they shift their weight just so they might even slide towards the wagon.