No, there is no drag-and-drop mechanic for survey question formats. For an example on how to implement such mechanics using regular <trial> etc. elements, see the "Tower of London" script available in the Task Library. But beware: The process is rather involved. Thus it might be advisable to consider some simpler alternative such as using basic drop-down elements or the like.
I would like to design a survey page with two boxes with a number of characters above in which participants have to drag the characters into one of the two boxes with their response recorded. Is this design possible? If not, is there any other kind of design that would do the job?