A system is disclosed for use in additively manufacturing an object. The system may have a reinforcement supply, a matrix supply, and a wetting module configured to separately receive reinforcement from the reinforcement supply and matrix from the matrix supply and to discharge the reinforcement wetted with the matrix. The wetting module may include a body forming an enclosure and having an inlet end configured to receive the reinforcement and an outlet end configured to discharge the reinforcement wetted with the matrix. The wetting module may also include a plurality of nozzles dividing the enclosure into at least a wetting chamber.