We investigate the intrinsic shapes of low-luminosity galaxies in the central 300 kpc of the Virgo cluster using deep imaging obtained as part of the Next Generation Virgo Cluster Survey ( NGVS ) . We build a sample of nearly 300 red-sequence cluster members in the yet unexplored -14 < M _ { g } < -8 magnitude range , and we measure their apparent axis ratios , q , through Sérsic fits to their two-dimensional light distribution–which is well described by a constant ellipticity parameter . The resulting distribution of apparent axis ratios is then fit by families of triaxial models with normally-distributed intrinsic ellipticities , E = 1 - C / A , and triaxialities , T = ( A ^ { 2 } - B ^ { 2 } ) / ( A ^ { 2 } - C ^ { 2 } ) . We develop a Bayesian framework to explore the posterior distribution of the model parameters , which allows us to work directly on discrete data , and to account for individual , surface brightness-dependent axis ratio uncertainties . For this population we infer a mean intrinsic ellipticity \overline { E } = 0.43 ^ { +0.02 } _ { -0.02 } , and a mean triaxiality \overline { T } = 0.16 ^ { +0.07 } _ { -0.06 } . This implies that faint Virgo galaxies are best described as a family of thick , nearly oblate spheroids with mean intrinsic axis ratios 1 : 0.94 : 0.57 . The core of Virgo lacks highly elongated low-luminosity galaxies , with 95 per cent of the population having q > 0.45 . We additionally attempt a study of the intrinsic shapes of Local Group ( LG ) satellites of similar luminosities . For the LG population we infer a slightly larger mean intrinsic ellipticity \overline { E } = 0.51 ^ { +0.07 } _ { -0.06 } , and the paucity of objects with round apparent shapes translates into more triaxial mean shapes , 1 : 0.76 : 0.49 . Numerical studies that follow the tidal evolution of satellites within LG-sized halos are in good agreement with the inferred shape distributions , but the mismatch for faint galaxies in Virgo highlights the need for more adequate simulations of this population in the cluster environment . We finally compare the intrinsic shapes of NGVS low-mass galaxies with samples of more massive quiescent systems , and with field , star-forming galaxies of similar luminosities . We find that the intrinsic flattening in this low-luminosity regime is almost independent of the environment in which the galaxy resides–but there is a hint that objects may be slightly rounder in denser environments . The comparable flattening distributions of low-luminosity galaxies that have experienced very different degrees of environmental effects suggests that internal processes are the main drivers of galaxy structure at low masses–with external mechanisms playing a secondary role .