I did find the 'right click attack' option to be a bit finnicky though. I tried to get it to attack some riflemen and it seemed to refuse to actually launch the rocket. Most likely it was because they were scrambling about and throwing off it's arc, but I did notice it being very particular in that firing mode.
You don't right-click to fire the rocket at a unit like you would do with any other unit where right is move/attack.
You select the sturmtiger, hit 'R' (or 'Z' if you are using grit-hotkeys) to select the firing ability, and you left-click on an enemy model. The rocket is an 'ability' that does not auto-fire when you right-click an enemy unit. For the Sturmtiger right-click is movement only.
Edit: So just to be clear, you always left-click to fire the rocket. Left-click on an enemy model and it will track, left-click on the ground and it will not track.