In a drive-thru Internet vehicular network, vehicles can communicate with access points (APs) for the Internet services. When a vehicle leaves the AP's transmission range, it encounters intermittent connectivity and unreliable connection. Using vehicle-to-vehicle (V2V) relay can loosen the limitation of transmission coverage. However, the harsh conditions of wireless channel and the rapid change of vehicle topology remain the challenge for relay selection. The relay candidate with a better channel quality can forward the packet with a higher data rate. Therefore, the MAC layer mechanisms are needed to exploit the multi-rate capability in the PHY layer. This paper utilizes the channel state information to exploit good channel quality for the relay link between the sender and the receiver. The objective is to increase the opportunity of channel access for the relay candidate with better channel quality. Considering various issues, such as the collision probability, packet delivery ratio, and delivery delay, we propose an analytical model to dynamically configure the window size for channel contention. Numerical results show that the proposed method outperforms the basic DCF and distance-based DCF.