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Formation of supermassive black holes (BHs) remains a theoretical challenge. In many models, especially beginning from stellar relic ‘seeds,’ this requires sustained super-Eddington accretion. While studies have shown BHs can violate the Eddington limit on accretion disc scales given sufficient ‘fuelling’ from larger scales, what remains unclear is whether or not BHs can actually capture sufficient gas from their surrounding interstellar medium (ISM). We explore this in a suite of multiphysics high-resolution simulations of BH growth in magnetized, star-forming dense gas complexes including dynamical stellar feedback from radiation, stellar mass-loss, and supernovae, exploring populations of seeds with masses $\sim 1\!-\!10^{4}\, \mathrm{M}_{\odot }$. In this initial study, we neglect feedback from the BHs: so this sets a strong upper limit to the accretion rates seeds can sustain. We show that stellar feedback plays a key role. Complexes with gravitational pressure/surface density below $\sim 10^{3}\, \mathrm{M}_{\odot }\, {\rm pc^{-2}}$ are disrupted with low star formation efficiencies so provide poor environments for BH growth. But in denser cloud complexes, early stellar feedback does not rapidly destroy the clouds but does generate strong shocks and dense clumps, allowing $\sim 1{{\ \rm per\ cent}}$ of randomly initialized seeds to encounter a dense clump with low relative velocity and produce runaway, hyper-Eddington accretion (growing by orders of magnitude). Remarkably, mass growth under these conditions is almost independent of initial BH mass, allowing rapid intermediate-mass black hole (IMBH) formation even for stellar-mass seeds. This defines a necessary (but perhaps not sufficient) set of criteria for runaway BH growth: we provide analytic estimates for the probability of runaway growth under different ISM conditions.