As interesting as speculation on who took the ticket book is, I think something that's too often overlooked is 'what for?'.
Like yes obviously, the winner gets a favor from a Family Head, but what would they ask MC for?
Most people don't know how powerless MC actually is, like the guy who wanted MCs help getting a lawyer for his parents.
If it's someone who doesn't know MCs situation, then stealing the book implies desperation, this is their one chance at a miracle even if they know cheating like this could anger the family they're about to ask a favor of. So chances are, whatever they ask for won't be malicious, or otherwise be anything that could damage MC even if in the end MC can't actually grant the wish.
But if it's someone who knows MCs situation, then the favor is likely to be of a much more political sort, and thus far more likely to be potentially malicious, or otherwise something that could damage MC should he grant it.
Rachel is often touted as a possible thief, so the interesting question is what would make her so desperate for a family heads favor that she would steal from a man she actively hates? What would make her desperate enough to trust him?
A family issue? But we haven't seen or heard anything about her family, not that this couldn't be how we're introduced to them.
Most of The Families are unlikely to be involved.
Sarah's family don't need to steal a favor, believing Sarah has him firmly on their said.
Hannah and Alison's family is headed by a man rigidly devoted to the greater good and honor. Unlikely to commission a theft.
Catherine's family aren't very politically motivated. Her father, the head, is a kind man and Catherine's brother the next head is supposed to take after him. Catherine's mother already jumped ship back to her brothers family.
Brian's family are the type of stoop to underhanded means, but I feel his mother would find stealing this book to gain a favor childish and beneath her. Besides it would be an acknowledgment of MCs status which she tried to prevent him gaining. Besides, the favor lottery was hardly secret, MC suddenly granting her a favor, would be too public a move for her.
I do think Brain himself, acting on his own could have gotten someone to steal the book for him. His favor would probably be something aimed at humiliating or otherwise harming MC