Is it possible for a distance-versus-time graph to be a vertical line

No because it would imply motion at infinent speed.

Yes, if something stays at the same distance for a prolonged period of time. Like imagine they were measuring the distance of the moon from the earth. If they measured it every night for 365 nights, you would have time increasing 1 day each day up to 365, but the distance would still remain the same (230,100 miles) each time you measured each day. So the graph would look like a vertical line.