What is the name of the character who got Rose involved in spying in Warlight?

The short answer to your question is that we are never really sure. It seems to be that the boy Rose cared for when she was a child, Marsh Felon (at least that's what she calls him), became a recruiter for British intelligence and introduced Rose to the world of espionage.
The central idea in Warlight, though, is that our memories are not static, that they shift and change based on our current situations and on knowledge we acquire as life goes on. Rose's son Nathaniel is our narrator (mostly), and as he matures, things about his parents' lives become much clearer in some cases and murkier in others.
The uncertainty that we, the readers, experience is intended to mirror at least a couple of things. First, we are meant to examine the "truth" of our own memories of the past by experiencing some of Nathaniel's confusion about his parents. He comes to realize that his and his sister's feelings of abandonment are probably misplaced. He begins to understand that they were much more closely watched and cared for by his parents' circle of friends, primarily the man they call "The Moth," while their parents were performing important work for the war effort.
Second, Ondaatje seems to want us to understand that even our ability to recall the truth of our past is hampered by our limitations as humans. We experience the world as it is happening and life does not proceed for most of us in an orderly way. It is chaotic and sometimes frightening. We are all proceeding from the incomplete information we have. Our memories are our way of attempting to order that which defies order.
So, yes, Marsh Felon is probably the person who introduces Rose to espionage, but the answer is much more complex than that. Rose's spying is a product of her relationship with her husband, with her sense of duty to the war effort and her belief that the best way to protect her children was to serve the country. Nathaniel and Rachel remember her much differently than as the person she truly was.