High-flux, high-repetition-rate neutron sources are of interest in studying neutron-induced damage processes in materials relevant to fusion, ultimately guiding designs for future fusion reactors. Existing and upcoming petawatt laser systems show great potential to fulfill this need. Here, we present a platform for producing laser-driven neutron beams based on a high-repetition-rate cryogenic liquid jet target and an adaptable stacked lithium and beryllium converter. Selected ion and neutron diagnostics enable monitoring of the key parameters of both beams. A first single-shot proof-of-principle experiment successfully implemented the presented platform at the Texas Petawatt Laser facility, achieving efficient generation of a forward-directed neutron beam. This work lays the foundation for future high-repetition-rate experiments towards pulsed, high-flux, fast neutron sources for radiation-induced effect studies relevant for fusion science and applications that require neutron beams with short pulse duration.