In writing this book, it appears that the author has deliberately held back from mentioning God or describing any religious activity. He probably does this as a literary device to heighten the fact that God controls and directs all the seemingly insignificant coincidences that make up the plot and result in deliverance for the Jews. This is particularly evident in chapter 6, the critical turning point in the story, where all sorts of "coincidental" events occur to bring the story to its happy ending.
The book of Esther becomes a convincing reminder of God's providential, behind the scenes, attention to every detail of life. Review the book of Ruth for a similar illustration of God's personal care, where in the midst of the turbulent period of the judges, the story of God's care for Ruth and Naomi testify to his ability and readiness to attend to our every need.
Do you believe God is working for your benefit in all things? (Romans 8:28)