Despite the fact that static simulations are unable, in many cases, to fully capture the complex seismic behavior of steel cylindrical storage tanks, the majority of previous studies on tank buckling have still been based on incremental static analyses. In this study, the effect of soil flexibility on the inelastic dynamic buckling of cylindrical steel storage tanks is investigated using Incremental Dynamic Analysis (IDA) and the Budiansky–Roth buckling criterion. Accordingly, a comprehensive dataset of the tank–foundation–soil system was generated by performing 800 nonlinear time-history analyses, and the sensitivity of elephant-foot buckling to soil flexibility was evaluated for two representative tanks, including a slender tank and a broad tank. The fragility analysis results indicate that, in the slender tank, the probability of elephant-foot buckling is sensitive to soil type. In particular, Soil Type 1 significantly increases this probability for peak ground acceleration levels below 0.4g. However, at higher seismic intensities, no significant differences were observed among different soil types, and the probability of buckling occurrence or exceedance becomes nearly identical. In contrast, the broad tank exhibits no significant sensitivity to variations in soil stiffness and flexibility and their fragility curves for different soil conditions were virtually the same for a wide range of the peak ground acceleration.