Why does Rover hold my customers' money for weeks in advance?

If your customers don't want to pay in advance, then tell them to book you closer to the actual date of service.

However, I do disagree with your assessment that this is "nothing like booking an airline ticket." It is exactly like booking a plane ticket. You want a seat on a certain date/time; therefore, you book it well in advance. Your credit card is charged immediately. If customers want a place for their pup and feel the need (especially for holiday periods) to get it settled in well in advance, then they need to pay for it.