A key unresolved question in microbial ecology is how the extraordinary diversity of microbiomes emerges from the interactions among their many functionally distinct populations. This process is driven in part by the cross-feeding networks that help to structure these systems, in which consumers use resources to fuel their metabolism, creating by-products which can be used by others in the community. Understanding the effects of cross-feeding presents a major challenge, as it creates complex interdependencies between populations which can be hard to untangle. We address this problem using the tools of network science to develop a structural microbial community model. Using methods from percolation theory, we identify feasible community states for cross-feeding network structures in which the needs of consumers are met by metabolite production across the community. We identify tipping points at which small changes in structure can cause the catastrophic collapse of cross-feeding networks and abrupt declines in microbial community diversity. Our results are an example of a well-defined tipping point in a complex ecological system and provide insight into the fundamental processes shaping microbiomes and their robustness. We further demonstrate this by considering how network attacks affect community diversity and apply our results to show how the apparent difficulty in culturing the microbial diversity emerges as an inherent property of their cross-feeding networks.