Timely updating of Internet of Things data is crucial for achieving immersion in vehicular metaverse services. However, challenges such as latency caused by massive data transmissions, privacy risks associated with user data, and computational burdens on metaverse service providers (MSPs) hinder the continuous collection of high-quality data. To address these challenges, we propose an immersion-aware model trading framework that enables efficient and privacy-preserving data provisioning through federated learning (FL). Specifically, we first develop a novel multi-dimensional evaluation metric for the immersion of models (IoM). The metric considers the freshness and accuracy of the local model, and the amount and potential value of raw training data. Building on the IoM, we design an incentive mechanism to encourage metaverse users (MUs) to participate in FL by providing local updates to MSPs under resource constraints. The trading interactions between MSPs and MUs are modeled as an equilibrium problem with equilibrium constraints (EPEC) to analyze and balance their costs and gains, where MSPs as leaders determine rewards, while MUs as followers optimize resource allocation. To ensure privacy and adapt to dynamic network conditions, we develop a distributed dynamic reward algorithm based on deep reinforcement learning, without acquiring any private information from MUs and other MSPs. Experimental results show that the proposed framework outperforms state-of-the-art benchmarks, achieving improvements in IoM of 38.3% and 37.2%, and reductions in training time to reach the target accuracy of 43.5% and 49.8%, on average, for the MNIST and GTSRB datasets, respectively. These findings validate the effectiveness of our approach in incentivizing MUs to contribute high-value local models to MSPs, providing a flexible and adaptive scheme for data provisioning in vehicular metaverse services.