Faster delivery, lower shipping costs, and a higher chance of product availability, are some of the benefits offered by an omnichannel business model. Assuming that ordering can only be done before the start of the selling season, we address two questions that are crucial for the success of pooling inventories. First, what is the optimal order-quantity decision for each channel, i.e., the brick-and-mortar store and the online channel, under different omnichannel fulfillment strategies? Second, who should get the sales credit? To answer these questions, we propose a noncooperative game theory approach and characterize the equilibrium order quantities in four scenarios, namely, no cross-fulfillment of customer's demand, ship-to-store, ship-from-store, and hybrid fulfillment. We compare the equilibrium strategies and outcomes, and obtain, among other things, that sales credit allocation is a key factor in determining whether the firm will benefit from the omnichannel strategies. We consider various sales credit allocation rules and provide insights on the best practice for each fulfillment strategy, taking into account customers' shopping preference.