All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
Stuttering does not prevent one from reciting the Noble Quran according to the rules of Tajweed. Bear in mind, dear questioner, that Tajweed is an act of the Sunnah. The Prophet, sallallaahu ‘alayhi wa sallam, used to recite the Noble Quran with measured recitation. It was narrated on the authority of Hafsah that the Prophet, sallallaahu ‘alayhi wa sallam, used to recite the Surah of the Noble Quran in such a slow-measured tone that it would become lengthier than the one longer than it. [Muslim]
Moreover, the Prophet, sallallaahu ‘alayhi wa sallam, praised the one who recites the Quran with Tajweed; as he said:
"The one who was devoted to the Quran will be asked to recite, ascend and recite carefully as he recited carefully when he was in the world, for he will reach his abode when he comes to the last verse he recites." [Ahmad, Abu Daawood and At-Tirmithi]
The correct method of treating this speech problem and lack of fluency is through practicing, continual training and reciting the Noble Quran slowly without haste. You should try to read each verse of the Surah letter by letter, outside the prayer, before you pray. Were you to be sure that you can recite these verses properly, then you should try reciting them before others who are able to recite the Noble Quran correctly. Allaah Willing, you will find this a successful treatment, and the most important thing is that you do not hasten while reciting the Noble Quran.
Imaam Al-Qurtubi elaborated on the interpretation of the verse that reads (what means): {…recite the Quran with measured recitation.}[Quran 73:4]: "(it) means: do not hasten in reciting the Quran; instead recite it slowly and clearly while pondering over its intended meanings."
Furthermore, Adh-Dhahhaak believed that it means to recite it letter by letter, whereas, Mujaahid said, “The one whom Allaah Loves most in reciting the Quran is the one who reads it in the slowest manner.”
If you did everything in your ability to eschew the mistakes in reciting the Quran, there would be no harm or blame on you after that. However, we advise you not to lead people in prayer unless you feel capable of avoiding the mistakes in recitation or stuttering so as to respect the status of the Imaam (who leads people in prayer) and to relieve yourself of hardship or blame as well.
However, if your case of stuttering is innate, you should not feel sad because of that and do not think badly of it; since there were great Muslim scholars of our righteous predecessors who suffered from the same speech problem, and it did not degrade their status by any means.