SMT solver bugs, including soundness issues, crashes, and assertion failures, can silently corrupt verification results. We present CAST, to our knowledge the first continuous fuzzing service for SMT solvers, integrating differential testing and type-aware mutation into CI pipelines with commit-aware corpus reduction. In a 90-day evaluation on CVC5 and Z3, CAST found 42 bugs (12 soundness issues), 33 confirmed and fixed upstream, none dismissed as invalid, with 99.2% workflow success rate over 14,006 workflow executions. Corpus reduction achieves 37.0% (CVC5) and 80.6% (Z3) average reduction while maintaining coverage of changed code. Continued operation has found 46 bugs to date.