Vision transformers (ViTs) are emerging with significantly improved accuracy in computer vision tasks. However, their complex architecture and enormous computation/storage demand impose urgent needs for new hardware accelerator design methodology. This work proposes an FPGA-aware automatic ViT acceleration framework based on the proposed mixed-scheme quantization. To the best of our knowledge, this is the first FPGA-based ViT acceleration framework exploring model quantization. Compared with state-of-the-art ViT quantization work (algorithmic approach only without hardware acceleration), our quantization achieves 0.47% to 1.36% higher Top-1 accuracy under the same bit-width. Compared with the 32-bit floating-point baseline FPGA accelerator, our accelerator achieves around 5.6x improvement on the frame rate (i.e., 56.8 FPS vs. 10.0 FPS) with 0.71% accuracy drop on ImageNet dataset for DeiT-base.