Magnetoresponsive polymers have gained increased attention in the design of soft actuators as they can be spatially as well as temporally activated and enable an external noninvasive control of movement. By introducing the magnetoresponsive properties in photocurable resins, one can fabricate personalized and complex structures (via vat photopolymerization 3D printing), whose movement can be conveniently controlled by an external magnetic field. Advancing from acrylate-based photopolymers, which often suffer from shrinkage stress, low monomer conversion, and oxygen inhibition, the fabrication of magnetoresponsive thiol-click photopolymers containing Fe3O4 nanoparticles as magnetic fillers is highlighted. The addition of the thiol crosslinker yields soft and flexible polymer composites, whose cure kinetics, viscosity, thermal, and mechanical properties are studied as a function of the thiol and filler content. Although cure rate and final monomer conversion decrease with rising filler concentration, the cure kinetics is reasonably fast at 6 wt%. The short pot life, a result of thiol-Michael reactions induced by Fe3O4 nanoparticles, and a high thiol content, are overcome by the addition of an appropriate stabilizer. As proof of concept, 3D structures are fabricated by digital light processing (DLP) 3D printing and their magnetically driven movement is demonstrated.