The fact that there is more documentation about William Shakespeare than any other figure in Elizabethan theatre shows that he had the fame and public prestige we would expect from someone who wrote the plays.
Absence of evidence is not evidence of absence. Just because there is no surviving documentation to show that he made his livelihood from writing does not mean that he didn't. It is possible there are just no surviving documents.
There is extensive evidence which places him in the theatre world. He was in the Lord Chamberlain’s Men, a troupe which would go on to become the King’s Men. But, perhaps most convincingly, his name is on the plays. Why would there need to be any further documentation beyond that?