The short answer is yes, chemical, biological, radiological, and/or nuclear attacks would likely be state-sponsored for the simple fact that rogue weapons of this nature are very difficult to come by. Highly sophisticated equipment, facilities, and skills are necessary to develop at least three of the four weapons. Even if a weapon was procured on the black market, it's worth noting that the developing nation would have to provide some guidance on its mechanics. If, for example, ISIS wanted a nuclear weapon, it would need a facility in which to develop one or to purchase one already developed. It would be almost impossible to develop their own with the eyes of the world on their every move. Therefore, it would be very difficult for a nation to wash its hands completely clean of a rogue weapon sold on the black market.
By contrast, whenever people are willing to sell their talents to the highest bidder, there is a possibility they will take their talents to nefarious actors. It is not always possible to connect the actions of these actors to a state. The best example is the group of Al-Qaeda members who weaponized plans on 9/11. 15 of the 19 terrorists were Saudi citizens, but there has been no evidence to date that Saudi Arabia sponsored their actions. Al-Qaeda instead found support from the Taliban in Afghanistan and, some evidence suggests, Pakistan. Finally, biological weapons are much easier to cultivate independently, and cheaply, than the other three categories. So, it may be possible for a non-state-sponsored organization to procure or design their own biological weapons. (However, we can still assume those weapons would have a much smaller reach than state-developed ones.)
Regarding the question of which state-sponsored program would be responsible, it would be difficult to say. North Korea seems an obvious answer, though they have not hid their nuclear ambitions behind rogue organizations thus far. Russia allegedly has a cache of weapons that remain unaccounted for since the fall of the Soviet Union; we know little about the chain of custody for these weapons. The bottom line is that the less accountable to the international community a state becomes, the more likely they would be to aide in state-sponsored terrorism.