Does the normal force require other forces to exist?

In this case, there wouldn't be a normal force between the blocks. Much like friction, the normal force only acts in opposition to other forces. From a logical real-world perspective, this makes sense too, since if the normal force acted between, say, two billiard balls sitting on a surface, they would start rolling away from each other, which isn't what we observe to happen.

Contrary to other answers so far, no force is needed to create a normal force. However, if there's no force counteracting the normal force, the situation is obviously not static. So in a static situation, there is always a force that balances the normal force.

Here's a brief explanation of the mechanism that causes the forces to be balanced in many real-life situations, and an example of a situation where a normal force exists without any other forces.

To start, we need to understand what makes a normal force appear.

Imagine putting a heavy box on a trampoline. First, the box pushes the trampoline down. But when the trampoline is stretched, the internal forces try to return it to its original position, which also pushes the box up.

This creates some oscillation, because after the trampoline is stretched too much, the restoring force is actually bigger than the gravity of the box. But eventually the energy dissipates and it settles to a point where the normal force balances the gravity of the box completely, and the box doesn't move.

Now consider a box lying on a floor. With a hard box on a hard floor, the oscillation is completely invisible as it's too small to see and dissipates fast. However, the main reason for the normal force is the same: The floor and the box get compressed a bit on the surface, and it is just enough to create a force that balances the gravity of the box.

If the normal force was bigger than the gravity of the box, the box would start rising, thus decreasing the compression of the floor, thus decreasing the normal force. If the normal force was smaller than the gravity of the box, the box would start going down, thus increasing the compression of the floor, thus increasing the normal force.

So in a static situation, the normal force must balance the force pushing the objects together. Simply because otherwise the situation wouldn't be static.

So now consider your situation. If there was a normal force between the two side-by-side boxes, they would push each other away. But after a very tiny movement, there wouldn't be any more compression, so there would be nothing to cause a normal force. So there is no normal force between the boxes in your situation. Simply because otherwise the situation wouldn't be static.

However, normal forces can appear without any other forces in a dynamic situation. Consider a gravityless environment, where a ball is flying towards a wall. When the ball and the wall touch, a normal force appears for the exact same reasons it appears in the box-on-a-floor or box-on-a-trampoline cases described above. However, there's nothing to balance the normal force: The ball and the wall would first compress, and then start decompressing, and all the time the normal force would be changing the momentum of the ball. Finally, the ball starts flying backwards and leaves the wall, and the normal force gets down to zero.