Uncovering the rules governing the nonequilibrium dynamics of the membranes that define biological cells is of central importance to understanding the physics of living systems. We theoretically and computationally investigate the behavior of model protocells-flexible quasispherical vesicles-that exchange membrane constituents, internal volume, and heat with an external reservoir. The excess chemical potential and osmotic pressure difference imposed by the reservoir act as generalized thermodynamic driving forces that modulate vesicle morphology. We identify an associated nonequilibrium morphological transition between a weakly driven regime, in which growing vesicles remain quasispherical, and a strongly driven regime, in which vesicles accommodate rapid membrane uptake by developing surface wrinkles. This transition emerges due to the renormalization of membrane mechanical properties by nonequilibrium driving. Further, using insights from stochastic thermodynamics we propose a minimal vesicle growth-shape law that remains robust even in strongly driven, far-from-equilibrium regimes.