Our research material comprised antivandal fire-retardant hybrid composites modified with inorganic and organic fillers intended for application in public transport vehicles. This paper presents an approach to studying their impact strength as a function of the composite thickness (3 to 6 mm) and striker shape (hemispherical, semicylindrical, wedge-shaped) used in the experimental stand. Group A composites, made of single fabric layers (n = 5), were thinner and their impact strength was lower by 73% than that for Group B composites made of double fabric layers. Study results show an almost threefold improvement in impact strength for a thickness increase of as little as 0.3 mm. Statistical analysis (the Shapiro–Wilk test, p > 0.05) did not show any significant differences in the quantitative evaluation of changes (n = 3) on the surface of the examined materials caused by impacts with strikers of different shapes. In turn, a linear correlation (Shapiro-Wilk test, W = 0.0857, p = 0.022) was found between impact strength and the thickness of the studied materials. It was observed that appropriate arrangement of fabrics and powder fillers can lead to a different distribution of forces and energy absorbed by the tested material. A lower impact strength was observed for the composite which had powder fillers in its composition, which caused the formation of microvoids in the structure of the material and thus led to a weakening in their strength properties. An effect of the placement of the glass fabric layer in the composite on the results was also observed. Moreover, SEM evaluation of the composites revealed their layered structure and the impregnation of woven fabrics with resin.