Does sleeping on the prayer rug after performing the night vigil prayer (qiyam al-layl) and supplicating for a specific need, and not waking up until a period of time has passed, indicate the prayer's acceptance or God's unwillingness to complete the supplication regarding this matter?

Your sleep during supplication has no bearing on its acceptance. However, there are signs that indicate the acceptance of a supplication, such as: humility (khashyah), weeping, and goosebumps. This may be followed by a tranquility of the heart, a calming of the spirit, and the emergence of energy and lightness, as if the supplication was a heavy burden that has been lifted.