A multivariate distribution can be described by a triangular transport map from the target distribution to a simple reference distribution. We propose Bayesian nonparametric inference on the transport map by modeling its components using Gaussian processes. This enables regularization and accounting for uncertainty in the map estimation, while still resulting in a closed-form invertible posterior map. We then focus on inferring the distribution of a spatial field from a small number of replicates. We develop specific transport-map priors that are highly flexible but shrink toward a Gaussian field with Matern-type covariance. The approach is scalable to high-dimensional fields due to data-dependent sparsity and parallel computations. We also discuss extensions, including Dirichlet process mixtures for marginal non-Gaussianity. We present numerical results to demonstrate the accuracy, scalability, and usefulness of our methods, including statistical emulation of non-Gaussian climate-model output.