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

Yes it is possible only if the x axis is distance and the y axis is time, providing the object being monitored or graphed never moves and you are utilizing a line graph for the assignment. time will increase (traveling vertically on the y axis or up/down on the graph) and distance will stay at a constant 0 (not traveling horizontally the x axis or left/right on the graph)

Not the way we generally draw graphs ... with time on the x-axis
and distance on the y-axis.

A vertical line would mean ...

-- The object is a every different distance all at the same time.

-- The object moves from one distance to another in zero time.
   That's the same thing as saying its speed is infinite.

Both of these are definite no-no's.  So you will never see the position
(distance) of a real moving object graphed as a vertical line.