"His father had stopped him from delivering groceries from the small store at the corner. This did not worry him at all. He picked the eyes out of every conversation, for he knew there were hundreds of other ways of getting a living. Once for fun, his father let him..."
I agree with Mirlo. it occurred to me just before I saw his reply. "blacking boots" is synonymous with "shining shoes". So his father allowed him to (let him) shine shoes (black boots) outside his house. Sorry for missing it.