Christian observances of Yom Kippur occur when a Christian-style Day of Atonement models itself on the Jewish holiday of Yom Kippur.
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Sabbatarian churches
As observed by the Living Church of God:
The Christian Day of Atonement is based on the English translation of the Jewish Holy day Yom Kippur. The day is commemorated with a 25-hour fast by Jews, but normally a 24-hour fast by Christians who observe it. While not observed by the mainstream of professing Christianity, the Christian groups (mostly those with origins in the old Worldwide Church of God) that do observe it usually refer to it as the Day of Atonement...
Generally, the Sacred Name Movement and groups reject Easter and Christmas as pagan in origin and observe the holy days of Leviticus 23 such as Yom Kippur, as well as Passover and the Feast of Weeks.
Messianic Jewish congregations
Messianic Jewish congregations devote serious effort at presenting a rationale for taking Yom Kippur. Such as by the Emmanuel Messianic Jewish Congregation (Clarksville, Maryland, USA):
For believers in Yeshua, both Jewish and non-Jewish, the observance of Yom Kippur can hold special significance. The repentance started at Rosh HaShanah comes to a culmination with atonement ten days later. As with the traditional Jewish community, those ten days (Yomim Nora'im) can take on spiritual meaning as we meditate on the meaning of the high holy days. Although there are not many customs directly relating to the ten days, the message could be applied to a believer's daily meditation at that time. Traditional readings from the book of Jonah, Hosea 14 and other pertinent passages can enhance one's appreciation of the season...Blessed be the Lord God, who has secured our salvation in Yeshua the Messiah! That is what Yom Kippur is all about for those who call on his name...
Jews for Jesus
Jews for Jesus describes its observances of this day as follows:
...Yom Kippur can be somewhat of a conundrum to Jewish believers in Y'shua. Do we fast and confess our sins like the rest of the Jewish community or do we rejoice in the knowledge that we're forgiven in Messiah? Many Jewish believers view Yom Kippur as a time for identification with our Jewish people, introspection for ourselves and intercession for loved ones, knowing all the while that Jesus is the One that makes us at one with God...
The same organization posts the histories of adherents who came to observe Yom Kippur as a part of worshiping Jesus (Yeshua or Y'shua):
Synagogue is hardly the scene to begin a story about believing in Jesus, but it was there my questions started...From that point on, the entire Bible opened up. God's own Word described the One who would come, die, bear the sins of humanity, and be rejected. I knew that I had found the answer to my prayers in Jesus. I had met the God of Israel. It was through this Yom Kippur prayer that I came to see myself before God. There is no explaining away of sin. But there is a sin-bearer and "…that through Him forgiveness of sins is proclaimed to you, and through Him everyone who believes is freed from all things, from which you could not be freed through the Law of Moses." (Acts 13:38-39)