To my knowledge of the National Constitution, there is not distinction between "active" and "alumni" members.  

It's true that once you are in APO (that is, paid your dues and gotten on the National List), you are in for life.  However, that does not necessarily mean you have a vote.

It boils down to your own Cast's Local Constitution/By-Laws.  If there is nothing there that states alumni can not vote, then there's where it stems from.  If you want this teacher/alumnus to not have a vote, you have to put it in the constitution.  

We had something close happen at my current chapter.  We held elections before initiations.  Pledges were allowed to be nominated for office.  Two were elected to office.  Some people made a stink about it, but it came down to the fact that there is nothing in our constitution that says that "can't" hold office.  

Hope that helps you with your problem.

Now and Forever!