Recent years have witnessed an upsurge of interest in employing flexible machine learning models for instrumental variable (IV) regression, but development of uncertainty quantification methodology is still lacking. In this work we present a novel quasi-Bayesian procedure for IV regression, building upon the recently developed kernelized IV models and the dual/minimax formulation of IV regression. We analyze the frequentist behavior of the proposed quasi-posterior, establishing minimax contraction rates in $L_2$ and Sobolev norms, and showing that the radii of its credible balls have the correct order of magnitude. We derive a scalable approximate inference algorithm, which has time cost comparable to the corresponding point estimation method, and can be further extended to work with neural network models. Empirical evaluation shows that our method produces informative uncertainty estimates on complex high-dimensional problems.