Q:

Is it legal and possible to trample someone with a controlled mount? 

In 5th edition, mounted combat with a controlled mount limits the actions of your mount into the Dodge, Dash, and Disengage actions.  However, what about charging at someone and trampling them?  Intuitively, it seems like it should be possible.  However, I can't find any rules to support this, or any to provide support for a reasonable ruling. 
There is a similar question but without an accepted answer of the more general case, in which it is asked if the mount can attack while being ridden.  This question is more specific in that I am asking if you can have the mount trample someone while you are controlling it (so it does not have its own initiative). 

A:

No, if you control your mount, it can't trample someone.  You said it yourself:

it has
  only three action options: Dash, Disengage, and Dodge. 

None of those includes trampling.  You might be thinking that you can trample enemies just by moving through them, but that's not how it works. 

You can move through a nonhostile creature’s space.  In
  contrast, you can move through a hostile creature’s space
  only if the creature is at least two sizes larger or smaller
  than you.  Remember that another creature’s space is
  difficult terrain for you. 

Moving through enemies doesn't do damage to them, or knock them prone, or anything else like that.  Here's how trampling works in D&D 5e:

Trampling Charge.  If the horse moves at least 20 feet straight
  toward a creature and then hits it with a hooves attack on
  the same turn, that target must succeed on a DC 14 Strength
  saving throw or be knocked prone.  If the target is prone, the
  horse can make another attack with its hooves against it as a
  bonus action. 

This particular example is taken from the Warhorse statblock, but many monsters (such as elephants and mammoths) have similar abilities.  Even a mount that has this ability won't be able to use it while being controlled, since it requires them to use their Hooves action, which they can't do.