Is there a minimum range for an Intercontinental Ballistic Missile? To be more concrete: Would it be possible to fire a ICBM Minuteman III from Malmstrom Air Force Base in Montana (USA) to hit Helena (the capital of Montana) about a hundred kilometers away?
If not, what is the minimal distance the missile needs to travel before hitting its target?
 A: I don't see any reason why a ballistic missile cannot hit an arbitrarily close point, unless there are such limitations in a specific design (say, software limitations). Moreover, it can hit a close point using a high trajectory or a low trajectory. If one uses a high trajectory, one should take into account the Earth's rotation. For example, if the missile speed is about 8 km/s, the high trajectory flight will take about 1600 s, and Helena, Montana, will move on the order of 400 km in 1600 s due to Earth's rotation, if I am not mistaken. @IamAStudent mentioned "a much sharper turn at the apex", but the missile would have low speed at the apex, so this should not be a problem.  
A: Yes..  It would be possible to hit something that close with an ICBM.   The system would have to be reprogrammed.  The warhead would enter into space and would travel around the globe before re-entering and hitting your target.  
This is a pretty good video of what happens on an ICBM launch.
https://www.youtube.com/watch?v=lOHHtIVP8Pw
A short-range ballistic missile (SRBM) is a ballistic missile with a range of about 1,000 kilometers (620 mi) or less. ... Within the U.S. Department of Defense, a medium-range missile is defined by having a maximum range of between 1,000 and 3,000 km and an ICBM is greater than 3500km.
So - to answer your question - could you launch a missile and have the warhead hit 100 miles away?  Yes, but it is not straight up and straight down.
