The reason why the apostles asked ‘who sinned’ is because they, like all Jews of the time, were influenced by the Mishnah. The Mishnah was an interpretation of the Old Testament. And it was oral (learnt by rote/memory). Later (after Jesus’s time) it was ‘written’ into what we now know as the Talmud.
Jesus often confronted the religious leaders of the day, where they would quote Mishnah, and Jesus would respond with Old Testament. Example...
MAT 15:1 Then the scribes and Pharisees who were from Jerusalem came to Jesus, saying, 2 ”Why do Your disciples transgress the tradition of the elders? For they do not wash their hands when they eat bread.”
The ‘tradition of the elders’ is referencing Mishnah. And Jesus response ...
MAT 15:3 He answered and said to them, “Why do you also transgress the commandment of God because of your tradition? 4 For God commanded, saying, ..... [snip]
And that is exactly what is happening in John 9
JOHN 9:2 And His disciples asked Him, saying, “Rabbi, who sinned, this man or his parents, that he was born blind?”
The Jews believed only by obeying the law could a person be blessed. This ‘belief’ comes straight out of Mishnah. And this persons parents were obviously not. [blessed] - because their son was born blind. That’s the only way that he could have been.
So, to your questions ....
“Was it a widely held belief among first century Jews that a man could have sinned before his birth in this way?” - yes, because of the oral traditions that the Jews had over them. (Mishnah)
“What could make the apostles say that?” - that was would have been a natural response - the obvious question to ask. They were very ‘intimate’ with those oral traditions - they were ‘part’ of many Jews.
Note they referenced Jesus as ‘rabbi’ - this indicates that this was early in their discipleship - and at this time, those asking the question only ‘saw’ Jesus as a ‘teacher’. It’s unlikely the question came from ‘the 12’. Jesus acquired many followers, but those closest to Him knew him as more than [just] a teacher.