It would have been very difficult for Northern civilians to get their hands on tobacco products during the war (in fact the Internal Revenue Act placed a tax on luxury items -- tobacco was included in that).

If you were a soldier, however, it was pretty easy to procure tobacco. Since most of the war was fought in the South you could easily help yourself to free tobacco at whichever plantation caught your eye and, as Northern and Southern soldiers did feel a certain kinship, there were occasional lulls in the fighting where soldiers would call a short truce to bury their dead and meet up to trade -- these trades included tobacco.

And finally, America is an expansive country and obviously, surveillance was not nearly at the level it is today, so some entrepreneurial people would smuggle tobacco into the north, as detailed here (this requires a JSTOR subscription).

So yeah, it was tricky, even if you were a soldier, but people were still able to get tobacco, it just required a bit more ingenuity and money than it did before or after the war.