We investigate the energy sources of random turbulent motions of ionized gas from H alpha emission in eight local star-forming galaxies from the Sydney-AAO Multi-object Integral field spectrograph (SAMI) Galaxy Survey. These galaxies satisfy strict pure star-forming selection criteria to avoid contamination from active galactic nuclei (AGNs) or strong shocks/outflows. Using the relatively high spatial and spectral resolution of SAMI, we find that - on sub-kpc scales, our galaxies display a flat distribution of ionized gas velocity dispersion as a function of star formation rate (SFR) surface density. A major fraction of our SAMI galaxies shows higher velocity dispersion than predictions by feedback-driven models, especially at the low SFR surface density end. Our results suggest that additional sources beyond star formation feedback contribute to driving random motions of the interstellar medium in star-forming galaxies. We speculate that gravity, galactic shear and/or magnetorotational instability may be additional driving sources of turbulence in these galaxies.