Oral cancer, a pervasive and rapidly growing malignant disease, poses a significant global health concern. Early and accurate diagnosis is pivotal for improving patient outcomes. Automatic diagnosis methods based on artificial intelligence have shown promising results in the oral cancer field, but the accuracy still needs to be improved for realistic diagnostic scenarios. Vision Transformers (ViT) have outperformed learning CNN models recently in many computer vision benchmark tasks. This study explores the effectiveness of the Vision Transformer and the Swin Transformer, two cutting-edge variants of the transformer architecture, for the mobile-based oral cancer image classification application. The pre-trained Swin transformer model achieved 88.7% accuracy in the binary classification task, outperforming the ViT model by 2.3%, while the conventional convolutional network model VGG19 and ResNet50 achieved 85.2% and 84.5% accuracy. Our experiments demonstrate that these transformer-based architectures outperform traditional convolutional neural networks in terms of oral cancer image classification, and underscore the potential of the ViT and the Swin Transformer in advancing the state of the art in oral cancer image analysis.