Is it possible for people with a stutter to have a career in academia? I understand that it is difficult for people with stuttering to verbally communicate and explain their ideas/research or to give lectures. Does that means there is no chance for them to work in academia even with the PhD degree?
I don't want to turn this into an answer about how universities or society in general should deal with disabilities.  I will offer what I have witnessed first hand through a good friend.  

Can you work in Academia?

Yes.  

Will people make fun of you?

Probably but who cares.  People make fun of others for everything.  

But...


don't try to hide your problem
embrace the stuttering.  Nobody cares that much unless you do.
spit out what you are trying to say.  The content and meaning are key.  Don't let the stuttering stop your speech.
don't try to dumb down your words to help the stuttering (to a point).  You will be judged by most on what you say, don't let stuttering get in the way of that.


So my friend - let's call him Paul (sorry Paul for using your real name) had a big banquet speech he was asked to do after much of his research was published.  Four of the faculty members/advisors Paul was thanking were named John...  So he says, "I would like to thank Joh-Joh-Joh-Joh-Joh-Joh-John.  (He catches his breathe and smiles at the crowd).  Well I had fou-fou-four Johns and I think I thanked too many."  It could have been two hours of drinking but people were honestly laughing for two minutes straight.  

In reality it probably would effect you more if you were in a field were public speaking were part of the field.  Paul was a scientist.  I hate to say this but I think the stuttering became his calling card and made him even more likable, and even better presenter.  

Add:  After seeing this question get a ton of hits I want to be clarify my answer.  Stuttering and other issues, whether it is a disability or just something that someone isn't good at - be honest with yourself about any "shortcomings" and don't make others feel awkward (I know this is hard).  Whether it is a strong accent, stuttering, talking low, saying "ummmm", whatever - if you know you have an issue own it and put your audience at ease.  

A perfect example of this is Jeff Foxworthy.  He readily admits that he talks like a redneck.  That is his shtick.  People like him and think he's hilarious because he openly talks about his accent and redneckedness.  If he came out to an audience, unknown to them, and tried to (poorly) hide his accent and redneck background, people would feel uncomfortable.   Everyone has issues, some worse than others.  Acknowledging your issues makes your more personable and relateable to your audience as they know that they too might have trouble talking in front of a large group.