What is the best place to make supplication during the Night Vigil Prayer (especially in the last third of the night): Is it during prostration, or in the Qunut supplication after completing its formulation, or by raising one's hands after finishing the entire prayer?

Supplicating during prostration is better, because it is one of the positions where prayers are answered, and because a servant is closest to his Lord when he is prostrating. A person can combine supplication during prostration with supplication after the Qunut and after concluding the prayer with Tasleem.