Additive technologies have been widely adopted in various industries. The choice of additive technology and material directly affects the functionality of the manufactured components. The development of materials with better mechanical properties has led to a growing interest in replacing traditional metal components with those manufactured using additive technologies. The application of Onyx as a material comes into consideration, which contains short carbon fibers to increase the mechanical properties. This study aims to experimentally verify the viability of substituting metal gripping elements with nylon and composite materials. The design of the jaws was customized to meet the requirements of a three-jaw chuck of a CNC machining center. The evaluation process involved monitoring the functionality and deformation effects on the clamped PTFE polymer material. When the metal jaws were applied, significant deformation of the clamped material occurred, which varied with the clamping pressure. This deformation was evidenced by the formation of spreading cracks on the clamped material and permanent shape changes in the tested material. Conversely, nylon and composite jaws manufactured using additive technology demonstrated functionality across all tested clamping pressures, without causing permanent deformation of the clamped material, unlike the traditional metal jaws. The results of this study confirm the applicability of the Onyx material and provide practical evidence of the potential for reducing deformation caused by clamping mechanisms.