It is sometimes confusing to me who should press "book it now." I feel like if I do it (as a sitter) I may appear presumptuous, pushy. But, the pet owner may feel the same way. They may think since the sitter hasn't done it, there may be something still in question.
I recently sent an enhancement request to allow the pet owner and sitter to click respective "I'm ready to book it" buttons. IMO, this might help indicate who's waiting on whom. What either person sees could be enhanced with prompts such as "The requester has indicated they're ready to book it. If you are, press 'Book it now.'" I think it would help guide the conversation. I could say "I notice you haven't checked off the 'I'm ready'. Is there something else I could help with? (Then it could more quickly lead to archiving the request -- which I also feel apprehensive about doing when I'm unsure whether they're waiting on me.).
In your case, since the dog owner has already said they'll drop off the dog, I would press "book it now" and let them know that they'll need to accept the booking before dropping off the dog.
BTW: I'm quickly discovering 80% of last-minute requests have other drama associated with them. I'm becoming more assertive about my expectations, limiting my exposure to wasted time. That's why I would be clear that the booking must be accepted prior to the dog being dropped off. (There have been reports of owners not booking. Then it turns into days of nagging them to book the stay which is already occurring.).