To walk we apply a force and the earth gives us a reaction. But the foot which get the force doesn't accelerate.
You can think of it this way: when you step forward, your body pushes backwards on your foot, I.e. accelerates it, but the ground pushes it forward (via friction) and the net acceleration of the foot is zero. Meanwhile, the equal and opposite reaction when your body pushes backwards on your foot is that your foot pushes forward on your body, so there is a net forward acceleration of your body. As well, nothing counteracts your foot's backward push on the ground, so the ground moves backwards because there is a net acceleration: but we tend to ignore that, because the ground is a massive planet and doesn't accelerate much from our human-sized forces.
The foot which gets the force is connected to the leg and the rest of the body*. Force applied to the foot is transmitted to the rest of the body. Furthermore, the force applied (by the earth) is a frictional force, and opposes any motion of the foot. In cases of low friction (walking on ice, for instance) the food does indeed move. The active force applied to the foot comes from the mass of the body and gravity.
*To quote the song: "Oh, the foot bone's connected to the ankle bone, the ankle bone's connected to the shin bone, the shin bone's connected to the knee bone, the knee bone's connected to the thigh bone, the thigh bone's connected to the hip bone, the hip bone's connected to the back bone, the back bone's connected to the neck bone, the neck bone's connected to the head bone. Now hear the word of the Lord."