The automotive industry is one of the biggest consumers of polymer composites. Aside from good mechanical properties, polymer composites have low mass, which positively affects the overall vehicle weight reduction and improves energy efficiency. Although polymer composites are used in various vehicle components, this paper focused on the application in vehicle bumper production. Two different composite plates with hybrid fiber layup were made; the first plate with a combination of glass and carbon fibers and the second with carbon and aramid. For comparison, and as a cheaper variant, a third plate was made only with glass fibers. In the first two plates, epoxy resin was used as the matrix, while in the third plate, polyester resin was used. Polyurethane memory foams of different densities (60, 80, 100 kg/m3) and thicknesses (10, 15, 20 mm) were used as impact force energy absorbers. With the factorial design of experiments, it was found that the thickness of the memory foam was the main influence factor. Without the use of memory foam, the hybrid composite, made of glass and carbon fibers, showed the highest energy absorption, while with the use of foam, the highest energy absorption was achieved with the glass fiber composite. Without the memory foam, the impact force measured on the glass/carbon hybrid composite was 9319.11 ± 93.18 N. Minimum impact force to the amount of 5143.19 ± 237.65 N was measured when the glass fiber composite plate was combined with the memory foam. When using memory foam, the impact force was reduced by 30–48%, depending on the type of composite used.