Vehicle platoons involve groups of vehicles travelling together at a constant inter-vehicle distance, with different common benefits such as increasing road efficiency and fuel saving. Vehicle platooning requires highly reliable wireless communications to keep the group structure and carry out coordinated maneuvers in a safe manner. Focusing on infrastructure-assisted cellular vehicle to anything (V2X) communications, the amount of control information to be exchanged between each platoon vehicle and the base station is a critical factor affecting the communication latency. This paper exploits the particular structure and characteristics of platooning to decrease the control information exchange necessary for the channel acquisition stage. More precisely, a scheme based on radio environment map (REM) reconstruction is proposed, where geo-localized received power values are available at only a subset of platoon vehicles, while large-scale channel parameters estimates for the rest of platoon members are provided through the application of spatial Ordinary Kriging (OK) interpolation. Distinctive features of the vehicle platooning use case are explored, such as the optimal patterns of vehicles within the platoon with available REM values for improving the quality of the reconstruction, the need for an accurate semivariogram modeling in OK, or the communication cost when establishing a centralized or a distributed architecture for achieving REM reconstruction. The evaluation results show that OK is able to reconstruct the REM in the platoon with acceptable mean squared estimation error, while reducing the control information for REM acquisition in up to 64% in the best-case scenario.