We propose an algorithm that performs joint target tracking and adaptive waveform selection at all the receivers of a multistatic radar system in a collaborative and fully distributed manner. The receivers track the target cooperatively utilizing an extended version of the recently-proposed diffusion Kalman filter and perform adaptive waveform selection by minimizing the mean-square tracking error. The centralized adaptive waveform-selection algorithms for multi-static radar systems require collection of the data measured by all the receivers in the system using possibly long receiver-transmitter communication links. On the other hand, the proposed algorithm solely relies on the local communications among the neighboring receivers to disseminate the information across the system. Moreover, all the receivers perform the same tasks of target tracking and optimal waveform selection and roughly converge to the same estimated values. Thus, the information of the best waveform to be transmitted as well as the target's track can be passed to the transmitter from any receiver. Simulation results show that the proposed algorithm outperforms the conventional fixed-waveform algorithms and performs close to the centralized adaptive waveform-selection algorithm.