Despite the development of more and more locally resonant (LR) structures of various resonant attachments, the application of LR structures in vibration attenuation is still limited by the widths of bandgaps. To achieve broadband vibration suppression at low frequencies, a LR plate with periodically attached stepped-frequency resonators (SFR) is proposed in this paper, where SFR refers to a group of local resonators with step-variable resonant frequencies. The extended plane wave expansion method is used to analyze the bandgap properties of the LR plate with SFR, and an exact series solution is used to analyze its vibration response. The merging of bandgaps based on SFR is revealed, and the parametric analysis shows that increasing the mass ratio or the damping of resonators is conducive to the merging effects, while increasing the frequency spacing makes the bandwidth of the merged bandgap to first increase and then decrease, as the bandgaps are gradually separated. Then, the optimal resonant frequencies of the SFR are obtained by maximizing the dominant vibration attenuation region. The numerical validation shows that the LR plate with SFR consisting of six resonators can provide broadband attenuation performance covering a frequency range from 190 to 450 Hz, which makes the LR structures more practical in vibration control. [ABSTRACT FROM AUTHOR]