PHNo, this is an example of a dangling participle. The subject of “waiting” should be the subject that immediately follows, “it.” But it isn’t. The subject of “waiting” never appears in the sentence at all, so we don’t know who is waiting.
It would be correct to say, “While I was waiting for the bus, it began to rain.” Or, “While he was waiting for the bus, it began to rain.” "Waiting" needs to have a clear subject.
It would be correct to say, “While I was waiting for the bus, it began to rain.” Or, “While he was waiting for the bus, it began to rain.” "Waiting" needs to have a clear subject.