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, the scenarios where the receiver is moving, are not addressed by recent studies. This paper tackles the angle of arrival (AOA) 3D-localization problem for multiple sparse signal sources with a moving receiver having limited antennas, potentially outnumbered by the sources. First, an energy detector algorithm is proposed to exploit the sparsity of the signal to eliminate the noisy samples of the signals. Subsequently, elevation and azimuth AOAs of sources are roughly estimated using two dimensional multiple signal classification (2D-MUSIC) method. Next, an algorithm is proposed to refine and estimate the elevation and azimuth AOAs more accurately. To this end, we propose a sparse recovery algorithm to exploit the sparsity feature of the signals. Then, we propose a phase smoothing algorithm to refine the estimations in the output of sparse recovery algorithm. Finally, K-SVD algorithm is employed to find the accurate elevation and azimuth AOAs of sources. For localization, a new multi-source 3D-localization algorithm is proposed to estimate the positions of sources in a sequence of time windows. Extensive simulations are carried out to demonstrate the effectiveness of the proposed framework.