Well I think warging is somewhat hereditary. Like Ursa says, we don't know enough about this power for absolutes. However, one can query the reason the Stark children were able to develop the warging ability was due to a combination of these three main circumstances.
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One, the prime requisite for warging: the Starks worship the old gods. The only known wargs are those who worship the old gods. Warging is a druidic ability, which requires a strong connection with nature and animals. Like any magic user, it can likely be taught, but one must first have an affinity for nature and the old gods.
Two, The Starks are of old blood, descended from the First Men who were more attuned with the old gods and consorted with the children of the forest. Thus, warging would come more natural to them. Halfhand specifically requested Jon to accompany him to the Frostfangs because he was descended from the First Men. There is power in the old blood of the First Men.
Three, the prime reason the Starkscan warg: the direwolves. These aren't normal wolves or dogs (man's best friend). These are DIRE wolves, unnatural creatures, seen only north of the Wall where the old gods still rule. The starks were able to develop a repore(sp?) with these wolves as they grew up. Being unnatural creatures, this repore would likely transcend a normal repore with a dog or cat. Bran was able to warg with Hodor only because he had expanded on his natural ability and had Jojen to help teach him.
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I believe all the children have the warging ability based on these three circumstances. After they diverge, their powers grow and weaken depending on their experiences. Since Robb and Rickon don't have POVs its difficult to ascertain whether they warg or not. But if you note, all the wolves exhibited personalities similar to their owners. Shaggydog, was impulsive, angry and aggressive, Lady, obedient and proper. As slik_rik said, there was a passage where a northman proclaimed Robb found a way past the Golden Tooth because his "wolf showed him the way". He probably had a similar experience that Jon did on the Frostfangs. I believe however he started ignoring these impulses after his marriage and his warging powers were abandoned. Sansa only had one pov before her wolf was killed, thus ending her repore, and her warging career before it even got started.
Anyways, sorry for the long post. I need to drink less!