This study investigated the use of non-formaldehyde binders in the production of plywood panels, focusing on mixtures containing 70% poly 4,4’-methylene diphenyl isocyanate (pMDI) and 30% soy flour (SF), along with blends of soy flour and agricultural residues (olive by-products—with and without extraction of their bioactive ingredients—and defatted hemp seeds). The basic properties of these biomaterials, such as moisture content, pH, and buffering capacity, were determined with laboratory analysis. Adhesive mixtures were characterized using attenuated total reflectance Fourier transform infrared spectroscopy (ATR-FTIR) and thermogravimetric analysis (TGA). The adhesive’s bonding ability was evaluated by manufacturing plywood panels on a laboratory scale, simulating industrial practices. The glue lines were visually inspected with a stereomicroscope. Micro-ATR-FTIR study of the cross-sections of plywood panels showed the full consumption of isocyanate groups indicating effective curing of the adhesive. Mixtures containing olive residues, particularly olive skin and stones, showed improved thermal stability in the TGA study. The mechanical properties of the plywood panels were assessed with three-point bending tests, while their shear strength and wood failure performance were tested according to the European standards used in the relevant industry (EN 314.1:2004 and EN 314.2:1993). In terms of flexural properties, the adhesive with non-extracted (NE) residual olive skin (ROS) showed the highest flexural strength of around 17 MPa and a flexural modulus of 650 MPa. The formulations containing extracted materials from hemp seeds (HSs) and residual olive skin (ROS) showed the best overall performance with wood failure values of 85% and 75% after the most severe cyclic test (EN314.1:2004-Pretreatment 5.1.3). Overall, the results showed that binders prepared with residual olive skin and defatted hemp seeds have promising performance and can be used in the manufacture of plywood panels.