By building kinetic models of biological networks one may advance the development of new modeling approaches while gaining insights into the biology. We focus here on building a stochastic kinetic model for the intracellular growth of vesicular stomatitis virus (VSV), a well-studied virus that encodes five genes. The essential network of VSV reactions creates challenges to stochastic simulation owing to (i) delayed reactions associated with transcription and genome replication, (ii) production of large numbers of intermediate proteins by translation, and (iii) the presence of highly reactive intermediates that rapidly fluctuate in their intracellular levels. We address these issues by developing a hybrid implementation of the model that combines a delayed stochastic simulation algorithm (DSSA) with Langevin equations to simulate the reactions that produce species in high numbers. Further, we employ a quasi-steady-state approximation (QSSA) to overcome the computational burden of small time steps caused by highly reactive species. The simulation is able to capture experimentally observed patterns of viral gene expression. Moreover, the simulation suggests that early levels of a low-abundance species, VSV L mRNA, play a key role in determining the production level of VSV genomes, transcripts, and proteins within an infected cell. Ultimately, these results suggest that stochastic gene expression contribute to the distribution of virus progeny yields from infected cells.