After his conquests in the Westerlands when King Robb returns to Riverrun, his first priority was retaking the North from Iron Islanders. His marriage to Jeyne Westerling had alienated the Freys and matters were further complicated by the treasonous murders committed by Karstarks. His passage to The North via The Twins was in peril.
As per Catelyn Chapter III in ASoS he was actively exploring alternate routes to go North, including through the Bloody Gate of Vale:
Robb: I've asked only that she (Lysa Arryn) open the Bloody Gate for us, and provide ships at Gulltown to take us north. The high road would be hard, but not so hard as fighting our way up the Neck. If I could land at White Harbor I could flank Moat Cailin and drive the ironmen from the North in half a year.
He was spending a lot of time brooding over the maps. Why did not he or any of his counsellors think of the following route:
- Follow the Red Fork from Riverrun to Darry.
- Reach White Harbour from the Bay of Crabs.
- Flank Moat Cailin from the rear and retake rest of the North.
Admittedly, it would have been a long sea journey and the only show stopper I can think of could be lack of sea-faring vessels at his disposal. But he only needed ships to carry his men and not war-ships. Certainly, the naval prowess of Riverlands at his disposal was sufficient for this voyage?
PS : I am still in the middle of ASoS, so beg your pardon if there is a reference of this option (and reasons it was not feasible) in subsequent chapters.