Equilibrium models for energy markets under uncertain demand and supply have attracted considerable attentions. This paper focuses on modelling crude oil market share under the COVID-19 pandemic using two-stage stochastic equilibrium. We describe the uncertainties in the demand and supply by random variables and provide two types of production decisions (here-and-now and wait-and-see). The here-and-now decision in the first stage does not depend on the outcome of random events to be revealed in the future and the wait-and-see decision in the second stage is allowed to depend on the random events in the future and adjust the feasibility of the here-and-now decision in rare unexpected scenarios such as those observed during the COVID-19 pandemic. We develop a fast algorithm to find a solution of the two-stage stochastic equilibrium. We show the robustness of the two-stage stochastic equilibrium model for forecasting the oil market share using the real market data from January 2019 to May 2020.