This text is part of the Covenant YHWH was making with his people. This covenant was met by their obedience to his word and statutes. By obeying YHWH, they would demonstrate their faith in His power to forgive - YHWH described how they ought to make atonement, and obedience to these statutes would have that effect.
"In this way he will make atonement for the Most Holy Place because of the uncleanness and rebellion of the Israelites, whatever their sins have been. He is to do the same for the tent of meeting, which is among them in the midst of their uncleanness. No one is to be in the tent of meeting from the time Aaron goes in to make atonement in the Most Holy Place until he comes out, having made atonement for himself, his household and the whole community of Israel." Leviticus 16:16-17
Similarly, the scapegoat passage that follows makes the assumption clear - if they obey YHWH in this way, their sins will be taken away:
"The goat will carry on itself all their sins to a remote place; and the man shall release it in the wilderness." Leviticus 16:22
These actions would be powerless without the word of YHWH - it is not that these specific actions held any power in and of themselves, but their efficacy lay in the Covenant that they fulfilled. The miraculous sign that signifies the effectiveness of the Law was the giving of the Law itself - YHWH had declared what they should do, and in order to receive it all that was left to them was to obey.
So no, there is no sign given in Leviticus 16, but the text is clear that the atonement was always understood to be successful, so no sign was necessary.
Addendum: Christians have traditionally understood this type of model to prefigure what would take place through Christ - the sign of forgiveness was the miraculous coming, death and resurrection of Jesus; meeting this sign/covenant with faith and baptism into Christ caused believers to receive atonement and birth into the new life made available by Jesus' sacrifice.
