Although influenza vaccine shortage is often attributed to low supply, it has been observed that even with abundant supply, a major shortage can still occur due to late delivery. In this paper, motivated by the influenza vaccine industry, we study a supply chain contracting problem in the presence of uncertainties surrounding design, delivery, and demand of influenza vaccine. In this supply chain, a manufacturer has insufficient incentive to initiate at-risk early production prior to the design freeze because it is a retailer who reaps the most benefits from selling more vaccines delivered on time. Anticipating that late delivery will lead to potential loss in demand, the retailer tends to reduce the order size, which further discourages the manufacturer from making an effort to improve its delivery performance. To break this negative feedback loop, a supply contract needs to achieve two objectives: incentivize at-risk early production and eliminate double marginalization. We find that two commonly observed supply contracts in practice, the Delivery-time-dependent Quantity Flexibility (D-QF) contract and the Late-Rebate (LR) contract, may fail to coordinate the supply chain because of the tension between these two objectives. To resolve such a tension, we construct a Buyback-and-Late-Rebate (BLR) contract and show that a properly designed BLR contract can not only coordinate the supply chain but also can provide full flexibility of profit division between members of the supply chain. Numerical experiments further demonstrate that the BLR contract significantly improves supply chain efficiency compared to the contracts used in the industry.