We present VRGaussianAvatar, an integrated system that enables real-time full-body 3D Gaussian Splatting (3DGS) avatars in virtual reality using only head-mounted display (HMD) tracking signals. The system adopts a parallel pipeline with a VR Frontend and a GA Backend. The VR Frontend uses inverse kinematics to estimate full-body pose and streams the resulting pose along with stereo camera parameters to the backend. The GA Backend stereoscopically renders a 3DGS avatar reconstructed from a single image. To improve stereo rendering efficiency, we introduce Binocular Batching, which jointly processes left and right eye views in a single batched pass to reduce redundant computation and support high-resolution VR displays. We evaluate VRGaussianAvatar with quantitative performance tests and a within-subject user study against image- and video-based mesh avatar baselines. Results show that VRGaussianAvatar sustains interactive VR performance and yields higher perceived appearance similarity, embodiment, and plausibility. Project page and source code are available at https://vrgaussianavatar.github.io.