Short answer: Nowhere to be found…and active everywhere.
One of the conundrums of the book of Esther is that there is no reference to God, prayer, worship, the synagogue, or any other evidence of interaction with the faith that the exiles brought with them from Jerusalem (or lack thereof since they were exiled because of idolatry). In fact there is a Greek addition to Esther which makes up for this obvious lack. It adds a lot of material, most of it having to do with worship and devotion, presumably because later readers were appalled at the lack of reference to God and wanted to “fix” things.
This omission is so obvious and stands out so clearly, that one must ask the question, “what was the author thinking?” which is exactly the question that it would seem, the author wanted us to ask. In other words, the lack of reference to the faith is a literary device that the author used on purpose. I don’t see how anyone could conclude anything else because the omission of reference to God is so glaring.
Why then? Karen Jobe, in her excellent commentary on the book of Esther is very helpful here. Her explanation is that the big question for exiles was: “Where are we at in terms of our faith here? We have been exiled because of idolatry, and now what is our relationship to God? Has he abandoned us? Is he still active? Does he love us?” This was the crucial question of the day for the exiles. While the answer seems obvious to us in hindsight, it would not have been so obvious to them at the time. Sure there was God’s promise that the exiles would return, and indeed some had returned 50 years before, but what about those who had remained in Babylon? Were they out of the Lord’s will and so abandoned by God, like the ten lost tribes of Israel?
Karen Jobes points out what we find from Esther: The great paradox of Esther is that God is omnipotently present even where God is most conspicuously absent.
This is what we find in Esther, and I think it is what the author of Esther wanted us to find.