We propose a model for network community detection using topological data analysis, a branch of modern data science that leverages theory from algebraic topology to statistical analysis and machine learning. Specifically, we use cellular sheaves, which relate local to global properties of various algebraic topological constructions, to propose three new algorithms for vertex clustering over networks to detect communities. We apply our algorithms to real social network data in numerical experiments and obtain near optimal results in terms of modularity. Our work is the first implementation of sheaves on real social network data and provides a solid proof-of-concept for future work using sheaves as tools to study complex systems captured by networks and simplicial complexes.