In the present study, the hydrogen embrittlement (HE) susceptibility of an additively manufactured (AM) 316L stainless steel (SS) was investigated. The materials were fabricated in the form of a lattice auxetic structure with three different strut thicknesses, 0.6, 1, and 1.4 mm, by the laser powder bed fusion technique at a volumetric energy of 70 J·mm−3. The effect of H charging on the strength and ductility of the lattice structures was evaluated by conducting tensile testing of the H-charged specimens at a slow strain rate of 4 × 10−5 s−1. Hydrogen was introduced to the specimens via electrochemical charging in an NaOH aqueous solution for 24 h at 80 °C before the tensile testing. The microstructure evolution of the H-charged materials was studied using the electron backscattered diffraction (EBSD) technique. The study revealed that the auxetic structures of the AM 316L-SS exhibited a slight reduction in mechanical properties after H charging. The tensile strength was slightly decreased regardless of the thickness. However, the ductility was significantly reduced with increasing thickness. For instance, the strength and uniform elongation of the auxetic structure of the 0.6 mm thick strut were 340 MPa and 17.4% before H charging, and 320 MPa and 16.7% after H charging, respectively. The corresponding values of the counterpart’s 1.4 mm thick strut were 550 MPa and 29% before H charging, and 523 MPa and 23.9% after H charging, respectively. The fractography of the fracture surfaces showed the impact of H charging, as cleavage fracture was a striking feature in H-charged materials. Furthermore, the mechanical twins were enhanced during tensile straining of the H-charged high-thickness material.