An important development in nonlinear dynamics is the discovery of chimera states that represent the coexistence of synchronized and desynchronized activity in populations of identically coupled oscillators. Identification and characterization of chimera states is currently an active area of theoretical and applied research. Here, we unveil a novel chimera state called "self-organized target wave chimera" in reaction-diffusion media where synchronized target waves spontaneously emerge from a pacemaker composed of asynchronous oscillators. This regime contrasts with a widely accepted perspective that synchronized target waves can be generated only by the individuals, which comprise the pacemaker, behaving in a synchronized manner. We characterize the features of self-organized target wave chimeras and present a phase diagram of existence of such a regime. The underlying phenomenological mechanism for the emergence of target wave chimeras is further analyzed. Our studies offer an affirmative answer to the existence of target wave chimeras in reaction-diffusion media and advance the general understanding of the onset of chimera states.