Magnons, namely spin waves, are collective spin excitations in ferromagnets, and their control through coupling with other excitations is a key technology for future hybrid spintronic devices. Although strong coupling has been demonstrated with microwave photonic structures, an alternative approach permitting high density integration and minimized electromagnetic crosstalk is required. Here we report a planar cavity magnomechanical system, where the cavity of surface acoustic waves enhances the spatial and spectral power density to thus implement magnon-phonon coupling at room temperature. Excitation of spin-wave resonance involves significant acoustic power absorption, whereas the collective spin motion reversely exerts a back-action force on the cavity dynamics. The cavity frequency and quality-factor are significantly modified by the back-action effect, and the resultant cooperativity exceeds unity, suggesting coherent interaction between magnons and phonons. The demonstration of a chip-scale magnomechanical system paves the way to the development of novel spin-acoustic technologies for classical and quantum applications.