Test-time adaptation (TTA) aims to improve model robustness under distribution shifts by adapting to unlabeled test data, but most existing methods rely on backpropagation (BP), which is computationally costly and incompatible with non-differentiable models such as quantized models, limiting practical deployment on numerous edge devices. Recent BP-free approaches alleviate overhead but remain either architecture-specific or limited in optimization capacity to handle high-dimensional models. We propose ZOTTA, a fully BP-free TTA framework that performs efficient adaptation using only forward passes via Zeroth-Order Optimization (ZOO). While ZOO is theoretically appealing, naive application leads to slow convergence under high-dimensional parameter spaces and unstable optimization due to the lack of labels. ZOTTA overcomes these challenges through 1) Distribution-Robust Layer Selection, which automatically identifies and freezes layers that already extract distribution-invariant features, updating only domain-sensitive layers to reduce the optimization dimensionality and accelerate convergence; 2) Spatial Feature Aggregation Alignment, which stabilizes ZOO by aligning globally aggregated spatial features between source and target to reduce gradient variance. Together, these components enable architecture-agnostic and stable BP-free adaptation. Extensive experiments on ImageNet-C/R/Sketch/A show that ZOTTA outperforms or matches BP-based methods, e.g., it reduces memory usage by 84% and improves accuracy by 3.9% over SAR on ImageNet-C.