We study the problem of embedding edgeless nodes such as users who newly enter the underlying network, while using graph neural networks (GNNs) widely studied for effective representation learning of graphs. Our study is motivated by the fact that GNNs cannot be straightforwardly adopted for our problem since message passing to such edgeless nodes having no connections is impossible. To tackle this challenge, we propose Edgeless-GNN, a novel inductive framework that enables GNNs to generate node embeddings even for edgeless nodes through unsupervised learning. Specifically, we start by constructing a proxy graph based on the similarity of node attributes as the GNN's computation graph defined by the underlying network. The known network structure is used to train model parameters, whereas a topology-aware loss function is established in such a way that our model judiciously learns the network structure by encoding positive, negative, and second-order relations between nodes. For the edgeless nodes, we inductively infer embeddings by expanding the computation graph. By evaluating the performance of various downstream machine learning tasks, we empirically demonstrate that Edgeless-GNN exhibits (a) superiority over state-of-the-art inductive network embedding methods for edgeless nodes, (b) effectiveness of our topology-aware loss function, (c) robustness to incomplete node attributes, and (d) a linear scaling with the graph size.