My hope is that the answer to this question is, "yes," because many people have a change of heart and realize how far they are from God when close to death. A sacrament is an outward sign of God's grace instituted by Christ, and who would need it more than a poor soul who has been away from the Church but wants to "Come Home" to the Church before hopefully going home to God? It seems to me that Anointing of the Sick would be such a blessing for someone who has been away from the Catholic faith... Am I anywhere close to the Church's teaching on this Father Pat?
Like most things of the Faith the answer is yes, but. Actually the Sacrament in question is called Anointing of the Sick which was formerly referred to as Extreme Unction. The expression Last Rites is not referred to in the ritual and should only be called such after a person has died. I have heard people say that they were given Last Rites several times during their lives and I always suggest that they were given the Sacrament of Anointing of the Sick perhaps with confession and Eucharist and prayers for the dying but since they did not die these were not their Last Rites. It is permissible for a non-practicing Catholic to receive Anointing of the Sick with Viaticum (Last Eucharist, literally means food for the journey) and prayers for the dying and the Apostolic Pardon BUT as long as they are not impenitent and contumaciously persisting in serious sin. If the person is unconcious and there is a doubt as to their repentance then the Sacrament may be given conditionally. It is always best to ask the person if they would like a priest to come, if they say yes then that is an admission of belief and their disposition towards the Sacrament. But not always. If they are also asked if they would like a wiccan priest or a shaman and they say yes then they are merely trying to cover all bases. The priest always determines if the person is disposed and even if the person is not responsive it is best to lean on the side of mercy and administer the sacrament and let God handle the rest. I remember anointing a man on Good Friday several years ago who was about to die and had not set foot in a Catholic Church for more than 30 years. Not only did he make a miraculous recovery, the greater miracle is that he is now a faithful weekly communicant and carries with him his Missal to Holy Mass.