We present LEGWORK (LISA Evolution and Gravitational Wave Orbit Kit), an open-source Python package for making predictions about stellar-origin gravitational-wave sources and their detectability in LISA or other space-based gravitational-wave detectors. LEGWORK can be used to evolve the orbits of sources due to gravitational-wave emission, calculate gravitational-wave strains (using post-Newtonian approximations), compute signal-to-noise ratios, and visualize the results. It can be applied to a variety of potential sources, including binaries consisting of white dwarfs, neutron stars, and black holes. Although we focus on double compact objects, in principle LEGWORK can be used for any system with a user-specified orbital evolution, such as those affected by a third object or gas drag. We optimized the package to make it efficient for use in population studies, which can contain tens of millions of sources. This paper describes the package and presents several potential use cases. We explain in detail the derivations of the expressions behind the package as well as identify and clarify some discrepancies currently present in the literature. We hope that LEGWORK will enable and accelerate future studies triggered by the rapidly growing interest in gravitational-wave sources.