It would be a surprising career choice! The final decision about the suitability of candidates to become Anglican priests rests with the bishop. There is strong precident that being an atheist would make one unsuitable.
However, there are cases of priests losing their faith. Some that come out as atheist are essentially sacked by the Church, so there is a strong disincentive to admitting you are an atheist if you're a priest. Also, priests that have had to resign after saying they were atheist have been contacted by many other priests who are also atheists "in secret" but who can't admit their lack of belief as they would lose their jobs.