We study compressive sensing in the spatial domain for target localization in terms of direction of arrival (DOA), using multiple-input multiple-output (MIMO) radar. A sparse localization framework is proposed for a MIMO array in which transmit/receive elements are placed at random. This allows to dramatically reduce the number of elements, while still attaining performance comparable to that of a filled (Nyquist) array. Leveraging properties of a (structured) random measurement matrix, we develop a novel bound on the coherence of the measurement matrix, and we obtain conditions under which the measurement matrix satisfies the so-called isotropy property. The coherence and isotropy concepts are used to establish respectively uniform and non-uniform recovery guarantees for target localization using spatial compressive sensing. In particular, nonuniform recovery is guaranteed if the number of degrees of freedom (the product of the number of transmit and receive elements MN) scales with K(log G)2, where K is the number of targets, and G is proportional to the array aperture and determines the angle resolution. The significance of the logarithmic dependence in G is that the proposed framework enables high resolution with a small number of MIMO radar elements. This is in contrast with a filled virtualMIMO array where the product MN scales linearly with G.