This work presents a novel type of locally resonant metamaterial structure conceived as a multi-beam system with internal local resonance. The proposed architecture consists of parallel beams coupled by periodic pairs of transverse stiffening elements, with compact internal blocks connected to each pair through slender beams. Motion of the internal blocks relative to the parallel beams induces internal resonance phenomena that generate wide band gaps in the dispersion diagram, enabling efficient attenuation of the flexural vibrations of the multi-beam system within targeted frequency ranges. Unlike conventional locally resonant configurations where resonators are externally attached, the proposed design integrates them within the structural layout, facilitating practical implementation in engineering applications. The investigation follows a combined analytical, numerical, and experimental approach. First, a simplified equivalent beam model is developed to capture the essential dynamic mechanisms governing wave propagation and to provide preliminary insight into band-gap formation. Subsequently, detailed three-dimensional finite element models are implemented using solid elements for in-depth numerical analyses. Wave propagation analysis of the infinite multi-beam system predicts significant local resonance band gaps, while frequency response analysis of the finite structure confirms strong vibration attenuation within the corresponding frequency intervals. To validate the numerical predictions, prototypes are fabricated through high-resolution resin 3D printing and subjected to experimental dynamic tests, which show very good agreement with the numerical predictions. Parametric analyses further demonstrate that the band-gap can be tuned by modifying the geometric properties of the slender beams connecting the internal blocks to the transverse stiffening elements, while damping contributes to widening the attenuation region without reducing in-band suppression performance. Overall, the study highlights the potential of the proposed locally resonant multi-beam system for advanced vibration control applications.