This paper proposes an integrated framework for coordinating multiple unmanned aerial vehicles (UAVs) in a distributed manner to persistently enclose and track a moving target without relying on external localization systems. The proposed framework consists of three modules: cooperative state estimators, circular formation pattern generators, and formation tracking controllers. In the cooperative state estimation module, a recursive least squares estimator (RLSE) for estimating the relative positions between UAVs is integrated with a distributed Kalman filter (DKF), enabling a persistent estimation of the target's state. When a UAV loses direct measurements of the target due to environmental occlusion, measurements from neighbors are aligned into the UAV's local frame to provide indirect measurements. The second module focuses on planning a desired circular formation pattern using a coupled oscillator model. This pattern ensures an even distribution of UAVs around a circle that encloses the moving target. The persistent excitation property of the circular formation is crucial for achieving convergence in the first module. Finally, a consensus-based formation controller is designed to enable multiple UAVs to asymptotically track the planned circular formation pattern while ensuring bounded control inputs. Theoretical analysis demonstrates that the proposed framework ensures asymptotic tracking of a target with constant velocity. For a target with varying velocity, the tracking error converges to a bounded region related to the target's maximum acceleration. Simulations and experiments validate the effectiveness of the proposed algorithm.