Is it permissible to resort to some shaykhs for treatment with the Qur’an for stuttering, knowing that medical treatment has not been effective?

Illness can be organic, or from the jinn (possession or magic), or the evil eye. The Quran is a healing for all ailments, so you should resort to it. We advise you to consult specialized doctors, and to go to someone who performs ruqyah with the Quran, supplications, and remembrances. It is best that you perform ruqyah on yourself, while ultimately relying on Allah.