Deciphering how quadrupeds coordinate their legs and other body parts, such as the trunk, head, and tail (i.e., body-limb coordination), can provide informative insights to improve legged robot mobility. In this study, we focused on sprawling locomotion of the salamander and aimed to understand the body-limb coordination mechanisms through mathematical modeling and simulations. The salamander is an amphibian that moves on the ground by coordinating the four legs with lateral body bending. It uses standing and traveling waves of lateral bending that depend on the velocity and stepping gait. However, the body-limb coordination mechanisms responsible for this flexible gait transition remain elusive. This paper presents a central-pattern-generator-based model to reproduce spontaneous gait transitions, including changes in bending patterns. The proposed model implements four feedback rules (feedback from limb-to-limb, limb-to-body, body-to-limb, and body-to-body) without assuming any inter-oscillator coupling. The interplay of the feedback rules establishes a self-organized body-limb coordination that enables the reproduction of the speed-dependent gait transitions of salamanders, as well as various gait patterns observed in sprawling quadruped animals. This suggests that sensory feedback plays an essential role in flexible body-limb coordination during sprawling quadruped locomotion.