It was definitely possible, though it wouldn't have been without considerable hassles, and a few favors cashed in along the way. So, I assume by the way OP phrased the question that they are referring to sanctioned crossings through enemy lines (and not stealth missions where permission wasn't granted). In this case, yes, it was possible for civilians to cross from the Confederacy into the Union (and vice versa). We definitely have some examples of this, too. What was needed in most cases was a pass (a letter, really), granting specific permission for one person to move through a military district or region. These would have been written personally by the military commander in charge of that area, or someone very near the top of the military pecking order in that region. Yet even this would have only helped a person get out of an area, and would have done only so much to get them into the OTHER "enemy" region. For example, British observer Arthur Fremantle arrived in Texas in March 1863, and worked his way all the way to Richmond in time to move with Lee's army towards Gettysburg. Strictly an observer, Fremantle had managed to get permission from various generals along the way to move freely through the Confederacy, until he got a pass from Robert E. Lee himself attesting to his neutral status (the note essentially promised that Fremantle was not a spy). This pass allowed Fremantle to travel north into Union lines to continue observing (though it wasn't without some suspicion...however, Lee's note actually got him pretty far). The fact that Fremantle had several notes from high ranking Confederate officers attesting to his neutral status (and the basic fact that he was British) surprisingly got him into the Union, so it shows that such movement was possible under the right circumstances and with the right people attesting for them.

On the flip side, Mary Todd Lincoln had a half-sister named Emilie Todd Helm, which made her Abe Lincoln's step half-sister. Emilie Helm's husband fought and died for the Confederacy, and Mary Todd asked Abe to grant Emilie a pass north so that the woman could visit for purely social reasons after the death of Helm's husband. Lincoln granted the request, and after the visit, wrote a pass for Emilie to go back south, which was honored. Obviously, if the president gives the okay, soldiers manning checkpoints are obliged to allow passage, so again, it wasn't without precedent for such a request to be granted.

This was not a normal occurrence, however, and would have required a high-ranking individual to sign the pass allowing movement, and some level of trust on the other side to recognize the validity of the enemy pass, and accept that the person was not a spy (so, a pass signed by Lincoln or Robert E. Lee would have been looked upon as fairly safe, as their "word" on the matter was seen as somewhat sacrosanct).

A normal person without political connections on the level of Lee or Lincoln would have needed a pass from the commanding general in the area to get out of that side's lines, and would have needed some pre-arranged agreement to get into the other side's lines (so a parlay would have needed to have been called, where it was established that a person was coming through, and agreements would have been made to allow it based on extenuating circumstances).

[Sources: Doris Kearns Goodwin, 'Team of Rivals'; Bruce Catton, 'Glory Road']