Please let me go pick up your sister at the station by car.
Literal
Younger-sister [object-を] station [up to-まで] car [with-で] meet [purpose-に] go [causative-させて] please.
~させてください ('please let me do X') is the causative + てください — a polite way to request permission to act. 迎えに行く ('go to pick up') is the purpose construction. 妹さん uses respectful さん when referring to someone else's younger sister, where you'd say plain 妹 for your own.