Accurately localizing multiple sources is a critical task with various applications in wireless communications, such as emergency services, including natural post-disaster search and rescue operations. However, scenarios where the receiver is moving have not been sufficiently addressed in recent studies. This paper tackles the angle of arrival (AOA) 3D-localization problem for multiple sparse signal sources with a moving receiver, which has a limited number of antennas that may be outnumbered by the sources. First, an energy detector algorithm is proposed to leverage signal sparsity for eliminating noisy samples. Subsequently, an iterative algorithm is developed to refine and estimate the AOAs accurately, initialized with previously estimated source locations and coarse elevation and azimuth AOAs obtained via the two-dimensional multiple signal classification (2D-MUSIC) method. To this end, we introduce a sparse recovery algorithm to exploit signal sparsity, followed by a phase smoothing algorithm to refine the estimates. The K-SVD algorithm is then applied to the smoothed output to accurately determine the elevation and azimuth AOAs of the sources. For localization, a new multi-source 3D-localization algorithm is proposed to estimate source positions using the refined AOA estimates over a sequence of time windows. Extensive simulations are carried out to demonstrate the effectiveness of the proposed framework.