Facial rigging - creating FACS-based blendshapes together with inner-mouth geometry (teeth, gums, and tongue) - remains a major bottleneck in 3D character production. Existing pipelines still require substantial designer effort, especially for manual landmark annotation, per-character template adjustment, and inner-mouth placement. We present OmniFaceRig, a fully automatic end-to-end pipeline that converts a static surface-only 3D character mesh, with no pre-modeled oral cavity, into an inner-mouth-aware FACS rig with up to 155 blendshapes, procedurally fitted teeth, gums, and tongue, and re-packed UV/texture. OmniFaceRig supports diverse topologies - humans, humanoids, long-muzzled animals (e.g., dogs, wolves, foxes), and short-muzzled animals (e.g., cats, bears, rabbits, tigers) - with no manual landmarks, no user-provided templates, and no per-asset setup. The pipeline combines hybrid VLM+CV riggability checking, multi-model face parsing, dense keypoint-driven template registration, procedural inner-mouth construction, and collision-aware blendshape transfer. For non-human characters, OmniFaceRig selects topology-specific face and inner-mouth templates and uses collision-aware inner-mouth fitting to reduce teeth-face intersections without exposing users to category-specific tuning. We also publicly release Omni-Bench, a freely available benchmark dataset of 1,000 biped 3D characters with FACS facial blendshapes and inner-mouth geometry, spanning humans, humanoids, cats, dogs, and other animals. Experiments show high final rigging success on screened Omni-Bench inputs, nearly complete face detection recall from the segmentation ensemble and reliable inner-mouth placement with low penetration. Together, OmniFaceRig provides an automatic path from static generated characters to animation-ready facial rigs across both human and non-human topologies.