Amazon prime is accelerating for anytime, so you are not mechaven for shabbos, and we have amira deamira from erev shabbos. the goy does it for his own benefit payed a fixed amount ahead of time should be mutar.
It’s muttar if the request can be fulfilled without doing melacha on Shabbos; normally assur if the request cannot be fulfilled without doing melacha on Shabbos (except in extenuating circumstances).
So for example, if you order something late Friday afternoon in the summer, so in order to deliver it by the 8 or 9pm promised delivery time, some melacha needs to be done on Shabbos, it would normally be prohibited.