Continuity mistake: After the priest writes down 666 on a piece of paper, he holds it upside down to reveal that it is 999. When we cut back to him a few seconds later, the numbers look completely different. (01:05:40)
Continuity mistake: In the church at the end when the windows are shattering, Arnie's got his right arm protecting his face in a wide shot and a closeup, then in the next shot (behind him) his left arm's up. we then cut in front of him again and his right arm's back up. (01:44:20)
Continuity mistake: When Satan enters the church demanding to know where Christine York is, one of the priests approaches him with a crucifix. The camera is facing the priest, Satan grabs the crucifix and stabs the priest right in the centre of his forehead (just above the eyebrow line). The camera angle changes to facing Satan and when the priest hits the deck, the crucifix is sticking out of his head higher up and above his left eyebrow. (01:22:30)
Continuity mistake: After the baby is born, the nurse puts a blue towel on her, completely covering her chest. The next shot, the baby's chest isn't covered. (00:06:00)
Continuity mistake: When Satan gets out of the car and turns up his collar, he has no gloves on his hands, but in the next shot as he is still turning up his collar, his hands are already gloved. (00:16:55)
Answer: The Vatican is aware of the birth, somewhere in the world, of Christine (though not knowing her identity at the time), and the Satanists are present at her birth and know that she is "the one." It's safe to assume that the Vatican, and the Catholic Church in general, has spies/informants in the Satanists' society, and were able, in the intervening 20 years between Christine's birth and the events of the film, to track her down to New York and send Aquinas there (the fact that he was sent by the Vatican is established).