What role does the fear of abandonment play in Willy’s life?
This is a good question. As you might imagine, fear of abandonment plays a very big role in Willy's life. First of all his father abandoned him. This has left a scar even as an adult. We can see this manifest in a few ways throughout the work. Let me give you a few examples.
First, Willy has this need to gain approval and be liked at all costs. While in moderation this desire is normal, but there is an unhealthy addiction to it. He is a veritable people pleaser. Why? The reason is because he is motivated by fear to gain approval. For example, when Ben comes to visit Willy's house, Willy wants Ben's approval of his life and sons. When Ben has to leave, Willy insists that he stay. It is the approval that Willy so desperately desires. The root is a fear of abandonment.
Second, we also see this in Willy's daydreaming, which is so vivid. In an alternate world, he is a different person, liked, and accepted. His escapism can be seen as another manifestation of his fear of abandonment.
Third, when Howard fires Willy, Willy takes it harder than is normal. The reason for this is because one of his greatest fears has been actualized - abandonment.
Fourth, the final straw is when Happy and Biff reject Willy. When this happens, Willy commits suicide. When he is abandoned, he has no recourse but to leave. Abandonment leads to further abandonment.