Muslims are meant to take the utmost care with the treatment of the Qur'an and in its recitation.
The proper recitation of the Quran is the subject of a separate discipline named tajwid which determines in detail how the Quran should be recited, how each individual syllable is to be pronounced, the need to pay attention to the places where there should be a pause, to elisions, where the pronunciation should be long or short, where letters should be sounded together and where they should be kept separate, etc. -- Wikipedia
However, this is a major obstacle to someone with a speech impediment (stuttering, lisps, etc.), who may be incapable of correct recitation.
Question: Can someone with a speech impediment recite the Qur'an?
When I say "can" above, I'm asking about permissibility (i.e., halal/haram) and desirability in the case where someone is incapable of accurately reciting the Qur'an due to some biological impediment. This is going to vary depending on the person's condition, and depending on how important it is for them to recite correctly (cf. an imam leading prayer vs. an everyday Muslim).
Obviously there's nobody physically stopping them picking up a Qur'an and reciting imperfectly, and moreover, reciting imperfectly will be part of the normal process of learning to recite the Qur'an.