In this letter, we report on unipolar vertical transport characteristics in c-plane GaN/AlGaN/GaN heterostructures. Vertical current in heterostructures with random alloy barriers was found to be independent of dislocation density and heterostructure barrier height. Percolation-based transport due to random alloy fluctuations in the ternary AlGaN is suggested as the dominant transport mechanism. This hypothesis is supported by simulations using drift-diffusion transport model incorporating statistical fluctuations of Al-composition and confirmed through experiments showing that non-random or digital AlGaN alloys and polarization-engineered binary GaN barriers can eliminate percolation transport and reduce leakage significantly. The understanding of vertical transport and methods for effective control proposed here will greatly impact III-nitride unipolar vertical devices.