I had my LBS put together a new frame with some old wheels. The new frame has wider dropouts. Should I be worried that the non-drive-side nut doesn't thread all the way onto the axle, or am I being overly paranoid? Not sure if this is worth taking it back to the LBS and making them fix it (thus wasting at least an hour, if not more, depending on if I have to come back another day).

What does the other side look like?

If there's more axle on the other side, they should have centered it for you.

There's a bit of axle protruding past the nut on the other side. but not a whole lot. If they centered it, both nuts would probably be pretty close to flush with the ends of the axles when fully tightened.

Unless you have a set of cone wrenches and can do it yourself, have them fix it. Considering the width of the rotating washer assembly on that track nut, there's most likely only about 4 threads engaged on the side pictured.