Aiming at the problems of complex calculation processes, insufficient risk data, and reliance on experts’ subjective judgments that exist in traditional probability analysis methods, this paper proposes a probability analysis method for hazardous chemical storage tank leakage accidents based on neural networks and fuzzy dynamic fault trees (Fuzzy DFT). This method combines fuzzy set theory (FST) and Bootstrap technology to accurately quantify the failure probabilities of basic events (BEs) and reduce the dependence on experts’ subjective judgments. Furthermore, an artificial neural network (ANN) model for tank failures is constructed. This model can accurately calculate the probability of tank leakage accidents by taking into account the dependency relationships among basic events. Finally, a long short-term memory (LSTM) network is utilized to analyze the dynamic evolution trend of the probability of storage tank accidents over time. In this paper, this method is applied to the case of the “11.28” Shenghua vinyl chloride leakage accident. The results show that the calculation results of this method are highly consistent with the actual situation of the accident, indicating that it is a scientific and effective method for analyzing the probability of hazardous chemical storage tank leakage accidents.