We present VISTA (Viewpoint-based Image selection with Semantic Task Awareness), an active exploration method for robots to plan informative trajectories that improve 3D map quality in areas most relevant for task completion. Given an open-vocabulary search instruction (e.g., "find a person"), VISTA enables a robot to explore its environment to search for the object of interest, while simultaneously building a real-time semantic 3D Gaussian Splatting reconstruction of the scene. The robot navigates its environment by planning receding-horizon trajectories that prioritize semantic similarity to the query and exploration of unseen regions of the environment. To evaluate trajectories, VISTA introduces a novel, efficient viewpoint-semantic coverage metric that quantifies both the geometric view diversity and task relevance in the 3D scene. On static datasets, our coverage metric outperforms state-of-the-art baselines, FisherRF and Bayes' Rays, in computation speed and reconstruction quality. In quadrotor hardware experiments, VISTA achieves 6x higher success rates in challenging maps, compared to baseline methods, while matching baseline performance in less challenging maps. Lastly, we show that VISTA is platform-agnostic by deploying it on a quadrotor drone and a Spot quadruped robot. Open-source code will be released upon acceptance of the paper.