Web attacks are one of the major and most persistent forms of cyber threats, which bring huge costs and losses to web application-based businesses. Various detection methods, such as signature-based, machine learning-based, and deep learning-based, have been proposed to identify web attacks. However, these methods either (1) heavily rely on accurate and complete rule design and feature engineering, which may not adapt to fast-evolving attacks, or (2) fail to estimate model uncertainty, which is essential to the trustworthiness of the prediction made by the model. In this study, we proposed an Uncertainty-aware Ensemble Deep Kernel Learning (UEDKL) model to detect web attacks from HTTP request payload data with the model uncertainty captured from the perspective of both data distribution and model parameters. The proposed UEDKL utilizes a deep kernel learning model to distinguish normal HTTP requests from different types of web attacks with model uncertainty estimated from data distribution perspective. Multiple deep kernel learning models were trained as base learners to capture the model uncertainty from model parameters perspective. An attention-based ensemble learning approach was designed to effectively integrate base learners' predictions and model uncertainty. We also proposed a new metric named High Uncertainty Ratio-F Score Curve to evaluate model uncertainty estimation. Experiments on BDCI and SRBH datasets demonstrated that the proposed UEDKL framework yields significant improvement in both web attack detection performance and uncertainty estimation quality compared to benchmark models.