Polyoxymethylene (POM) is one of the most popular thermoplastic polymers used in the industry. Therefore, the interest in its potential applications in rapid prototyping is understandable. Nevertheless, its low dimensional stability causes the warping of 3D prints, limiting its applications. This research aimed to evaluate the effects of POM modification with ethylene-vinyl acetate (EVA) (2.5, 5.0, and 7.5 wt.%) on its processing (by melt flow index), structure (by X-ray microcomputed tomography), and properties (by static tensile tests, surface resistance, contact angle measurements, differential scanning calorimetry, and thermogravimetric analysis), as well as very rarely analyzed emissions of volatile organic compounds (VOCs) (by headspace analysis). Performed modifications decreased stiffness and strength of the material, simultaneously enhancing its ductility, which simultaneously increased the toughness even by more than 50% for 7.5 wt.% EVA loading. Such an effect was related to an improved linear flow rate resulting in a lack of defects inside the samples. The decrease of the melting temperature and the slight increase of thermal stability after the addition of EVA broadened the processing window for 3D printing. The 3D printing trials on two different printers showed that the addition of EVA copolymer increased the possibility of a successful print without defects, giving space for further development.