Their intended destination is the Amazon, where they intend to hunt jaguar. They've sent their guns and other supplies separately and are taking a somewhat leisurely boat ride from America to the Amazon when Rainsford falls overboard. One can assume Whitney continues on to the destination or the nearest port to arrange to find Rainsford; we never hear of Whitney again, so if he did search for Rainsford, he likely avoided the island due to its reputation or a simple belief that Rainsford would not have made it there swimming. It's possible that Whitney does arrive on the island to find Rainsford after the "Game" ends, especially given the fact that no one is aware of Rainsford's mishap until, we can assume, morning, so they wouldn't know where exactly he fell overboard or where to begin looking. A rescue search could span many miles and take many days before it is given up. Regardless, we do not know how or even if Rainsford leaves the island, so we definitely do not know if he and Whitney are reunited.