You can impute and this will be fine. The missing data is missing completely at random (experementor error) so there is no problem here (even listwise deletion would be unbiased here though you would have substantial less power).
Look here
http://sites.stat.psu.edu/~jls/mifaq.html. For more detail look at the books by LIttle and Rubin or Craig Enders.
EDIT: The only thing maybe that would make the data missing at random is that there is a time or ordering effect here. As latter individuals got the question and earlier people did not. This is easily dealt with by creating a variable relating to the order (or date/time/whatever is appropriate) in which the participants completed the questionnaire. You then use this variable in your missing data model.