Yes, you can neck rein in a snaffle. There are a couple of approaches.
The style taught by the military, frequently seen in polo and other higher speed riding is described by George Morris thus:
"Both hands move over in the direction of turn, which causes the outside rein to press against and even cross the horse's neck. It is not surprising that this rein aid is a commonly used western control, for it demands a prompt turn of the animal's shoulders by its bearing action, hence the name "bearing" rein. I do not teach this or advocate this rein aid for normal hunter or equitation riding, but restrict it to sharp turns in a jumper time class where the emphasis lies on speed and instantaneous response..." Hunt Seat Equitation
The US Cavalry manual described it thus for a left turn:
"The right hand is carried just over the crest of the neck, and acts toward the left front. The rein, to be effective, should bear against the right side of the upper half of the neck, as this part of the neck is more sensitive to the rein than that near the shoulders. It is an artificial effect, and not powerful, but is the one habitually used with trained horses, to change direction without changing speed, particularly in polo. By using the left opening rein in early training, and later combining the right bearing rein with it, obedience to the right bearing rein is easily taught....The rider's legs normally remain in place, acting only to sustain the gait."
That is the way I like to do it. In the picture below, Bandit is using a solid shank curb bit, but he acts the same in a snaffle. The rein is pressing against the right side of his neck as he turns left. The horse does not get confused at the pulling on the right side of the mouth because horses don't analyze bits. They seek release, and Bandit knows what is expected of him - so his head is turned left and his feet are moving us to the left, doing a fairly tight 180 turn.
FWIW, he responds well to neck reining like this even if afraid or in tight quarters.
A different approach is used with some western riding, using a lot more slack in the reins. The video below is the one I used to teach my horses neck reining, even though he doesn't like the approach I use. The way he teaches it, though, still works. The second video makes a good point about teaching a horse neck reining: