“Our Father, Our King” Prayer on Shabbat

We do not say Avinu Malkeinu on Shabbat, because most of the prayer is asking G-d to save us from distressing situations. On Shabbat, we are to minimize thinking about distressing things, so we generally do not even pray about them. However, if the need is immediate, such as a person who is gravely sick, we would pray for him even on Shabbat.
Therefore, on Yom Kippur that coincides with Shabbat, at the Neiilah (Last) Prayer, the need for salvation is immediate, so we say Avinu Malkeinu even though it is Shabbat.