The Local Mutual Exclusion (LME) problem is a variant of classical Mutual Exclusion (ME) problem and can be considered as an extension of dining philosopher problem. In LME, no two neighboring nodes can enter the critical section (CS) simultaneously, whereas two non-neighboring nodes can be in their CS simultaneously. The resource allocation problem in Flying Ad hoc Networks (FANETs), is relatively an unexplored area despite having several potential applications. The present paper proposes LME problem for FANETs and provides a leader-based algorithm named as Request Collector Local Mutual Exclusion (RCLME) for the same. To the best of our information, LME problem is introduced first time in Flying Ad hoc Networks. The striking feature of the proposed algorithm is the introduction of a fuzzy logic-based leader election that considers the node speed, node direction, link quality, and the distance from the resource. The correctness proof of the RCLME algorithm has been presented. The simulation results show that RCLME algorithm significantly outperforms other related algorithms available in the literature; specially, when the number of nodes is large. The use of fuzzy logic and request collector improves the efficiency, fault tolerating capacity and ability to handle volatility.