It's impossible to ship foodstuffs by land anyway. Not beyond fifty miles and even that is stretching it. The damn draw animals consume more than they can pull over these distances. An oxen would go through its cargo in about ten days - and makes about eight miles per day.
Prior to the railway, the only way to ship bulk products was literally shipping it.
Their primary ports are fortified. Indeed, they are below their castles.
Any invader would have fun besieging those. Nothing like surviving frost in thin tents or trying to place a ladder upon icy surfaces while the defenders enjoy their warm fires and sufficient winter storages.
The Vale exports food anyway. They don't need the ports.
But it isn't reasonable. You write like people would actually be equal in Westeros. They aren't. Some people are worth more than others and they get all the benefits of their high station.
And a Stark bastard is pretty much at the top of the list of these people. There may be two hundred people more privileged than Jon in entire Westeros. In the NW, only Benjen himself.
Ned Stark cares about his daughters and their wolves. Cat Stark did. Robb Stark did. Bran Stark did. Sansa Stark did. Arya Stark did. And at least four of them would write Jon and have the means to have it delivered.