The subject must be motionless for conventional radar-based non-contact vital signs measurements. Additionally, the measurement range is limited by the design of the radar module itself. Although the accuracy of measurements has been improving, the prospects for their application could have been faster to develop. This paper proposed a novel radar-based adaptive tracking method for measuring the heart rate of the moving monitored person. The radar module is fixed on a circular plate and driven by stepping motors to rotate it. In order to protect the user’s privacy, the method uses radar signal processing to detect the subject’s position to control a stepping motor that adjusts the radar’s measurement range. The results of the fixed-route experiments revealed that when the subject was moving at a speed of 0.5 m/s, the mean values of RMSE for heart rate measurements were all below 2.85 beat per minute (bpm), and when moving at a speed of 1 m/s, they were all below 4.05 bpm. When subjects walked at random routes and speeds, the RMSE of the measurements were all below 6.85 bpm, with a mean value of 4.35 bpm. The average RR interval time of the reconstructed heartbeat signal was highly correlated with the electrocardiography (ECG) data, with a correlation coefficient of 0.9905. In addition, this study not only evaluated the potential effect of arm swing (more normal walking motion) on heart rate measurement but also demonstrated the ability of the proposed method to measure heart rate in a multiple-people scenario.