This paper aims to implement a simultaneous localization and mapping (SLAM) algorithm for a bionic robotic dolphin in underwater simulation environment. Through combining UWSim with ROS, an underwater simulation environment is constructed through importing various environmental models including coral, tropical fish, gullies, and so on. Thereafter, as an autonomous platform for underwater SLAM, a bionic robotic dolphin with its dynamic model is particularly built in the environment to perform dolphin-like swimming. Based on the sensor information from an onboard camera and an inertial measurement unit, the VINS-Mono algorithm is employed and implemented by the robotic dolphin to position itself underwater. Extensive simulations are conducted to validate the effectiveness of the employed SLAM algorithm in various dolphin-like motions of the robotic dolphin, such as linear, circular, and spiral motions. The obtained results provide valuable guidance for the underwater SLAM research.