Agricultural cargo vehicles are responsible for applying severe soil pressures. However, the ground straw cover can attenuate the loads applied by wheels to the soil surface. This research evaluated the effect of three tires, p1—Radial Very Flex, p2—Radial Improved Flex, and a p3—Bias Ply tire, on three amounts of straw on the soil surface (0, 15, and 30 Mg ha−1). We adopted a completely randomized design (CRD) with a rigid surface for three replications for the total contact area and punctual area claws. The soil bin test verified the deformable surface, tread marks, and soil penetration resistance (SPR). The tire’s claw design determines its punctual contact area, and the construction model determines the total contact area. The contact area in the soil bin increased linearly due to a increase in straw covering, reducing sinkage; p2 to 30 Mg ha−1 straw shows the most significant contact area, p1 and p3 showed no difference. A straw increase from 0 to 30 Mg ha−1 increased the contact areas by 25.5, 38.0, and 20.0% for p1, p2, and p3, respectively. Compared to the rigid surface, the p1 and p3 contact areas in the soil bin increased 6.2, 6.8, and 7.8 times in bare soil, 15, and 30 Mg ha−1; for p2, this increase was up to 4.2, 4.5, and 5.9 times on the same surfaces. Keeping the straw on the soil improves its physical quality by reducing the SPR, so the straw has a buffer function in the wheel–soil relationship.