Principal-agent problems study contracts for goods or services that a principal (seller) should offer an agent (buyer). The goal is for the principal to optimize the quantity and price in the contract offered to an agent with uncertain demand, where the principal has estimated a distribution for the agent's demand. The agent's demand distribution can be discrete or continuous. A deterministic optimization solution to the discrete distribution problem delivers a contract with price and quantity options targeted towards each possible demand realization. When the demand distribution is continuous, the optimal contract becomes a continuous function of the demand space. This paper introduces a sample average approximation to the continuous distribution problem using methods for solving the discrete distribution problem. We explore using numerical results an example motivated by carbon capture and storage systems.