Context. Cosmic voids are a promising probe of cosmology for spectroscopic galaxy surveys due to their unique response to cosmological parameters. Their combination with other probes promises to break parameter degeneracies. Aims. Due to simplifying assumptions, analytical models for void statistics represent only a subset of the full void population. We present a set of neural-based emulators for void summary statistics of watershed voids, which retain more information about the full void population than simplified analytical models. Methods. We built emulators for the void size function and void density profiles traced by the halo number density using the Q UIJOTE suite of simulations that spans a wide range of the Λ cold dark matter (ΛCDM) parameter space. The emulators replace the computation of these statistics from computationally expensive cosmological simulations. We demonstrate the cosmological constraining power of voids using our emulators, which offer orders-of-magnitude acceleration in parameter estimation, capture more cosmological information compared to analytical models, and produce more realistic posteriors compared to Fisher forecasts. Results. In this Q UIJOTE setup, we recover the parameters Ω m and σ 8 to within 14.4% and 8.4% accuracy, respectively, using void density profiles. Incorporating additional information from the void size function improves the accuracy for σ 8 to 6.8%. We demonstrate the robustness of our approach with respect to two important variables in the underlying simulations: the resolution and the inclusion of baryons. We find that our pipeline is robust to variations in resolution, and we show that the posteriors derived from the emulated void statistics are unaffected by the inclusion of baryons in the Magneticum hydrodynamic simulations. This opens up the possibility of a baryon-independent probe of the large-scale structure.