While vision transformers (ViTs) have shown consistent progress in computer vision, deploying them for real-time decision-making scenarios (<1 ms) is challenging. Current computing platforms like CPUs, GPUs, or FPGA-based solutions struggle to meet this deterministic low-latency real-time requirement, even with quantized ViT models. Some approaches use pruning or sparsity to reduce the model size and latency, but this often results in accuracy loss. To address the aforementioned constraints, in this work, we propose EQ-ViT, an end-to-end acceleration framework with the novel algorithm and architecture co-design features to enable the real-time ViT acceleration on the AMD Versal adaptive compute acceleration platform (ACAP). The contributions are four-fold. First, we perform in-depth kernel-level performance profiling and analysis and explain the bottlenecks for the existing acceleration solutions on GPU, FPGA, and ACAP. Second, on the hardware level, we introduce a new spatial and heterogeneous accelerator architecture, the EQ-ViT architecture. This architecture leverages the heterogeneous features of ACAP, where both FPGA and artificial intelligence engines (AIEs) coexist on the same system-on-chip (SoC). Third, On the algorithm level, we create a comprehensive quantization-aware training strategy, the EQ-ViT algorithm. This strategy concurrently quantizes both the weights and activations into 8-bit integers, aiming to improve the accuracy rather than compromise it during quantization. Notably, the method also quantizes nonlinear functions for efficient hardware implementation. Fourth, we design the EQ-ViT automation framework to implement the EQ-ViT architecture for four different ViT applications on the AMD Versal ACAP VCK190 board, achieving accuracy improvement with 2.4%, and average speedups of 315.0, 3.39, 3.38, 14.92, 59.5, and $13.1\times $ over computing solutions of Intel Xeon 8375C vCPU, Nvidia A10G, A100, Jetson AGX Orin GPUs, AMD ZCU102, and U250 FPGAs. The energy efficiency gains are 62.2, 15.33, 12.82, 13.31, 13.5, and $21.9\times $ .