To what extent is Willy the anti-hero?

I wouldn't call Willy an anti-hero. While there is some ambiguity around the term, an antihero is generally a character who undergoes the classic hero's journey without having all of the typical moral qualities of a conventional hero. This is a term that is more applicable to epic than to tragedy. In tragedy it is much more normal for the hero of the story to be flawed. Being flawed in a tragedy does not make the protagonist an antihero. Given this, though I do agree that Willy is the primary subject of the play, I would call him a tragic hero rather than an antihero. 

He follows more in the tradition of Greek tragedy. In this genre of drama a largely good-natured person suffers due to a crucial flaw. Willy is certainly more flawed than the typical greek hero, but I would argue he is essentially still good at the core. Despite his cruel behavior, cheating, and obsession with success, he is in many ways a victim of a vapid society, a poor upbringing, and a misconception of the drivers of true happiness. Deep down he still loves his sons and wants them to do well in life. His is blinded by vanity and insecurity.

An antihero is an extraordinary person who accomplishes extraordinary things despite less than perfect moral qualities. Willy is in many ways the opposite of this. He is an ordinary man who accomplishes very little, despite a core desire to be good. He falls short because of his flaws and his failures.