We propose a novel distributed user-car matching method based on a contract between users to mitigate the imbalance problem between vehicle distribution and demand in free-floating car sharing. Previous regulation methods involved an incentive system based on the predictions of origin-destination (OD) demand obtained from past usage history. However, the difficulty these methods have in obtaining accurate data limits their applicability. To overcome this drawback, we introduce contract-based coordination among drop-off and pick-up users in which an auction is conducted for drop-off users' intended drop-off locations. We theoretically analyze the proposed method regarding the upper bound of its efficiency. We also compare it with a baseline method and non-regulation scenario on a free-floating car-sharing simulator. The experimental results show that the proposed method achieves a higher social surplus than the existing method.