Both are required for walking$^*$.
You need friction to accelerate when you want to start walking, stop walking, change speeds while walking, etc. This is because you need a horizonal force to change your horizontal speed. This force is friction. It arises due to interactions between your feet and the ground you walk on. Therefore, by Newton's third law, the ground is pushed on by friction in the opposite direction of your horizonal acceleration.
However, don't discount the normal force. It is a vertical force (on level ground). Therefore this force is what keeps you from accelerating downward into the ground due to gravity. It also is one of the factors in determining how strong the previously mentioned friction force can be. A larger normal force typically means a larger possible friction force before sliding between your feet and the ground occurs. Therefore, without the normal force you wouldn't be able to walk either.
$^*$ Of course other forces like gravity, internal forces in your body, etc. are also important for walking. The physics of walking can get pretty complex. However you just asked about these two forces (friction and normal force), so I will just focus on those two.