We demonstrate quantum many-body state reconstruction from experimental data generated by a programmable quantum simulator by means of a neural-network model incorporating known experimental errors. Specifically, we extract restricted Boltzmann machine wave functions from data produced by a Rydberg quantum simulator with eight and nine atoms in a single measurement basis and apply a novel regularization technique to mitigate the effects of measurement errors in the training data. Reconstructions of modest complexity are able to capture one- and two-body observables not accessible to experimentalists, as well as more sophisticated observables such as the Rényi mutual information. Our results open the door to integration of machine learning architectures with intermediate-scale quantum hardware.