Abstract State-of-the-art spectral energy distribution (SED) analyses use a Bayesian framework to infer the physical properties of galaxies from observed photometry or spectra. They require sampling from a high-dimensional space of SED model parameters and take >10–100 CPU hr per galaxy, which renders them practically infeasible for analyzing the billions of galaxies that will be observed by upcoming galaxy surveys (e.g., the Dark Energy Spectroscopic Instrument, the Prime Focus Spectrograph, the Vera C. Rubin Observatory, the James Webb Space Telescope, and the Roman Space Telescope). In this work, we present an alternative scalable approach to rigorous Bayesian inference using Amortized Neural Posterior Estimation (ANPE). ANPE is a simulation-based inference method that employs neural networks to estimate posterior probability distributions over the full range of observations. Once trained, it requires no additional model evaluations to estimate the posterior. We present, and publicly release, SEDflow, an ANPE method for producing the posteriors of the recent Hahn et al. SED model from optical photometry and redshift. SEDflow takes ∼1 s per galaxy to obtain the posterior distributions of 12 model parameters, all of which are in excellent agreement with traditional Markov Chain Monte Carlo sampling results. We also apply SEDflow to 33,884 galaxies in the NASA–Sloan Atlas and publicly release their posteriors.