Being honest, If I wanted to achieve this, I would get myself an open source quad-copter, and adjust the control system to apply some "Input Constraints". Basically allowing the built in stabilization system to keep the drone stable and level, while only allowing me to modify the desired altitude. Depending on how advanced the system is, it should also be able to maintain its position in the world frame and eliminate any drift that occurs. You will not be able to create a drone that only flies up and down by removing parts from a standard quad-copter, as these sources of thrust are relied upon to keep the quad-copter stable, which is a deceptively complicated task.
To your question of "Could you enable a drone to fly vertically with only one propeller?", to which the answer is yes, but good luck with sorting out the rotational forces from the propeller without some other source of thrust to cancel it out.
As for drones that only fly vertically, sure, there are a few. The most obvious is a Hot Air balloon, if you decide to build your own and automate it. A cheap R/C Helicopter may also do the trick (From memory, some of the old cheap ones can only fly up/down, and rotate on the spot), but they can also be quite unstable, and certainly won't maintain their horizontal position.
If you are willing to stretch your definitions a bit, a Rocket kind of counts, but you probably shouldn't make one of those.