Can one state prohibit it's citizens from buying goods that carry specific taxes making it cheaper elsewhere?

Tobacco is a special case. Interstate transportation of tobacco is subject to joint state and federal regulation. Each state has the power to prohibit sales of cigarettes in its state unless the package bears a revenue stamp. It also has the right to prohibit introduction into the State for personal consumption of unstamped tobacco.

Most tobacco for personal consumption represents an economic trickle too small to be worth paying attention to. However, if a state locates a tobacco superstore right next to the border with a higher-rate neighboring State, the flow of traffic might be worth skimming from time to time.

Even other products are sometimes monitored this way. New York sales tax agents used to prowl shopping malls in Delaware, note the license plate numbers, and send proposed bills for the compensating use tax (the difference in sales tax is collectible by the destination state).

Just because it's interstate commerce does not mean that it can't be regulated by the states. To answer your question, the answer is no, it can't prohibit buying the cigarettes, but it can prohibit bringing them back into the home state.

No, state A cannot prohibit people from buying things in state B. However, in the case of tobacco or alcohol, it can prohibit bringing the goods back home. For other products, it can impose a "use tax" equal to the difference in sales tax between the two states.

Your question has also found application in the area of the buying of liquor in a state, where the tax may be low, and bringing it back into your home state. In the city where I'm from, this happened 24/7. It was against the law (in the state with the higher tax) but it couldn't be stopped. The law enforcement officials in the state (where the liquor was cheaper) wouldn't allow the activity to be curtailed.

No but what they do is make it illegal to bring them back across the state line. According to the laws in the states I know of cigarettes have to have the tax stamp from the state you're in.

No, you can not prohibit someone from buying something. That would be illegal and it would set you up to be prosecuted for limiting the right to liberty, which is the essential law of our country. About the only thing you could do with that is to pass a nationwide general tax - which would be possible, sense being it's not something essential to life.

No, but state B can limit the number of those items allowed to cross the state line at one time, say only 2 cartons of cigarettes allowed to leave North Carolina per vehicle, per trip.