The correct way to repunctuate the sentence in the structural style is the following one:
The strange boy whistled and put his hand into his pocket as far as the big coat sleeves would let them go.
Explanation:
"The strange boy whistled and put his hand into his pocket" is a quick sequence of actions, and as so it needs no pause, that is, no comma, let alone a semicolon.
"as far as the big coat sleeves would let them go" is an adverb clause, and adverb clauses that appear at the end of a sentence do not have to be preceded by a comma.
