We present Splat-MOVER, a modular robotics stack for open-vocabulary robotic manipulation, which leverages the editability of Gaussian Splatting (GSplat) scene representations to enable multi-stage manipulation tasks. Splat-MOVER consists of: (i) $\textit{ASK-Splat}$, a GSplat representation that distills latent codes for language semantics and grasp affordance into the 3D scene. ASK-Splat enables geometric, semantic, and affordance understanding of 3D scenes, which is critical for many robotics tasks; (ii) $\textit{SEE-Splat}$, a real-time scene-editing module using 3D semantic masking and infilling to visualize the motions of objects that result from robot interactions in the real-world. SEE-Splat creates a "digital twin" of the evolving environment throughout the manipulation task; and (iii) $\textit{Grasp-Splat}$, a grasp generation module that uses ASK-Splat and SEE-Splat to propose candidate grasps for open-world objects. ASK-Splat is trained in real-time from RGB images in a brief scanning phase prior to operation, while SEE-Splat and Grasp-Splat run in real-time during operation. We demonstrate the superior performance of Splat-MOVER in hardware experiments on a Kinova robot compared to two recent baselines in four single-stage, open-vocabulary manipulation tasks, as well as in four multi-stage manipulation tasks using the edited scene to reflect scene changes due to prior manipulation stages, which is not possible with the existing baselines. Code for this project and a link to the project page will be made available soon.