When should one supplicate during the night vigil (qiyam al-layl)? Should it be during the prayer or after it? And if it is during the prayer, should it be after saying "Sami'a Allahu liman hamidah" (Allah hears the one who praises Him) or during prostration?

There is no specific time for supplication during Qiyam al-Layl (night vigil prayer). It is permissible before and after the prayer, and it is preferred during the time of Sahar (the last part of the night before dawn), based on the Almighty's saying: (And in the hours before dawn, they would ask forgiveness) and the Hadith: "Our Lord, the Blessed and Exalted, descends every night to the lowest heaven when only the last third of the night remains, and says: 'Who is calling upon Me that I may answer him?'" It is permissible to supplicate with any worldly or otherworldly supplication, as long as it does not involve sin or severance of kinship ties. Supplications from the Sunnah (Ma'thur) are preferred. Supplication is prescribed during every pillar of the prayer, and it is preferred during prostration (sujud) due to the Hadith: "The closest a servant is to his Lord, the Mighty and Exalted, is when he is prostrating, so increase your supplications." Also, after the Tashahhud and before the Tasleem (salutation). As for rising from Rukoo' (bowing), no specific supplication has been reported for it; rather, one says: "Sami'a Allahu liman hamidah" (Allah hears those who praise Him), followed by "Rabbana wa lakal-hamd" (Our Lord, and to You be all praise).