Neural posterior estimation (NPE) is a simulation-based approach to Bayesian inference that trains a neural network to approximate the posterior distribution from simulated parameter - data pairs, bypassing likelihood evaluation. We apply NPE -- to our knowledge for the first time -- to stochastic susceptible-infectious-removed (SIR) epidemic models observed through final outcome data, considering both homogeneously mixing and household-structured populations. Such data arise naturally in retrospective outbreak investigations and household transmission studies, yet inference is computationally challenging: data-augmentation Markov chain Monte Carlo (MCMC) can be slow to mix in large populations and difficult to implement, while Approximate Bayesian Computation (ABC) suffers from low acceptance rates, particularly for large populations or unlikely outcomes. The discrete, low-dimensional nature of such observations makes this setting particularly well suited to NPE. We show that a logNormal posterior approximation, parameterised by a feed-forward neural network, accurately recovers reference posteriors across a range of population sizes and transmission regimes, and extends naturally to joint inference on global and local transmission rates in the household model. Once trained, the network produces approximate posterior distributions in seconds and generalises reliably to population sizes and structures not seen during training. Performance on both synthetic and real outbreak datasets is consistently strong, with results in close agreement with published analyses.