We propose a theoretical framework for training Graph Neural Networks (GNNs) on large input graphs via training on small, fixed-size sampled subgraphs. This framework is applicable to a wide range of models, including popular sampling-based GNNs, such as GraphSAGE and FastGCN. Leveraging the theory of graph local limits, we prove that, under mild assumptions, parameters learned from training sampling-based GNNs on small samples of a large input graph are within an $ε$-neighborhood of the outcome of training the same architecture on the whole graph. We derive bounds on the number of samples, the size of the graph, and the training steps required as a function of $ε$. Our results give a novel theoretical understanding for using sampling in training GNNs. They also suggest that by training GNNs on small samples of the input graph, practitioners can identify and select the best models, hyperparameters, and sampling algorithms more efficiently. We empirically illustrate our results on a node classification task on large citation graphs, observing that sampling-based GNNs trained on local subgraphs 12$\times$ smaller than the original graph achieve comparable performance to those trained on the input graph.