This work presents a new strategy to virtual bidding based on distributionally robust optimization (DRO) using a Wasserstein distance. Virtual bidding, a mechanism used in two-settlement electricity markets, allows participants to arbitrage price differences between day-ahead and real-time markets. Traditional optimization methods for virtual bidding often rely on precise probabilistic models of market behaviour, which are unavailable in practice due to the inherent complexity and volatility of electricity markets. To tackle these challenges, this work formulates the virtual bidding problem as a DRO problem, incorporating conditional value at risk (CVaR) into the objective to manage downside risk under volatile conditions. Tractable reformulations which can be efficiently solved to optimality are provided. The proposed strategy is developed and tuned using a 12-month training set to identify optimal parameters. The strategy is evaluated on historical pricing data from the New York Independent System Operator (NYISO) on an 8-month testing set. The results show improved performance over benchmarks, achieving higher Sharpe and Calmar ratios, as well as increased profit per MWh. Through this DRO framework, a more reliable virtual bidding strategy that enhances profitability while effectively managing risk in uncertain market environments is presented.