Is it permissible for a person suffering from a chronic, incurable disease to perform Umrah with the intention of seeking a cure, and what is the proper way to supplicate to Allah for that?

It is recommended to turn to Allah and supplicate to Him to remove what afflicts you, to consult doctors, and to take the necessary means. There is no specific supplication for this condition, but the door of supplication is wide, so one can supplicate for healing, for the improvement of one's condition, for ease in affairs, and for the relief of distress.