During the period of hostilities from the 1950's until 1975, there was never a US ground attack into North Vietnam presumably due to escalation and other strategic objectives. The South Vietnamese leadership, in its various stages, often operated separately from American interests despite its dependence.
Did the South Vietnamese government or the ARVN invade or attempt to invade North Vietnam whether in opposition to or by secret sanction of American interests?
No, they did not and there are two important reasons for this.
The first is that South Vietnam did not want to seem as an aggressor. North Vietnam, time and time again, denied any connections (beyond support) with the FNL and tried to make it out to be a grassroots movement in South Vietnam instead of being an insurgency movement completely orchestrated and led by the North Vietnamese. In time, of course, North Vietnam themselves started to send their own troops to support the main force VC units that now existed and won battles against the ARVN. By invading North Vietnam, the South would not only give up their claim of being victims of communist aggression and face further international criticism, but also an expansion of the war which the South simply was not prepared or capable of doing - not to mention the risk of triggering a Soviet or, more likely, a Chinese direct military intervention.
On the other hand, right from the start, the ARVN military doctrine was that of defense. It was trained specifically to repel a North Vietnamese invasion which at the time was believed to be imminent. This not only explains the reasons as to why the ARVN lacked any sort of inadequate response against an insurgency in their own back garden. With this in mind, not to mention the severe problems that deteriorated the ARVN throughout the war: corruption, demoralized and poorly trained troops (albeit valiant ones!) and dismal leadership - an invasion of the North on its own behalf would not have led to much but the cost of which would be tremendous. As we can see by the Easter Offensive in 1972, even the well-trained and well-motivated conventional PAVN troops from the North couldn't fully succeed with a conventional invasion of South Vietnam. It would be a whole other story three years later but by then, they were not the same armies that had fought against each other in 1972.
South Vietnam, with US assistance, did stage some raids on North Vietnamese targets as part of Operational Plan 34A. This secret program involved the covert insertion of South Vietnamese commandos (initially by land, though later missions were amphibious landings using swift boats) to raid North Vietnamese bases and gather intelligence.
In fact, the USS Maddox is thought to have been helping screen an OPLAN 34A mission when it was attacked by North Vietnamese ships in the infamous Tonkin Gulf Incident.