I wonder if you have entered your entire question. The reason I wonder is because, In “Contents of the Dead Man’s Pocket,” Tom does not actually permit himself to look down. The narrator specifically says that he did not do this. So I wonder if your question was supposed to ask why he does not permit himself to look down. However, he later looks down without meaning to do so, and you may be asking about that. In this answer, I will discuss why he would not permit himself to look down and why he actually did look down. Hopefully, one of those will be the answer to your actual question.
When Tom first goes out on the ledge, he shuffles along without looking down. In fact, we are told that
He simply did not permit himself to look down, though the compulsion to do so never left him; nor did he allow himself actually to think.
Tom does not want to permit himself to look down or to think about anything because that would distract him from what he is trying to do. He told himself earlier that he would be able to stay on the ledge indefinitely if it were only a few feet off the ground, so he needs to prevent himself from dwelling on the fact that he is actually 11 stories up. Later on, we see why this is important. When Tom accidentally looks down, he completely loses his composure. He is so frightened by being so far up that he comes very close to fainting and falling. Because he wanted to avoid this, Tom did not permit himself to look down.
So why did Tom look down if he was not permitting himself to do so? The answer is that he did it accidentally while trying to reach the paper. Remember, the ledge was probably about a foot wide and the paper was down on the ledge. Because the ledge was narrow, with the wall right in front of it, Tom could not just bend down and pick up the paper. Instead, he had to try to reach down in such a way that he could touch the ledge without losing his balance. To do this, he had to duck his head and lower his right shoulder, and that allowed him to accidentally look down and see how far up he was.
So, if you are asking why he did not permit himself to look down, it was because he did not want to lose composure and possibly fall. If you are asking why he did end up looking down, it is because he accidentally looked down while contorting his body to try to reach the paper.