This is in reply to your 5/14 post What did the hight priest do during those years? (fivedoves.com) in which you asked the following questions:

What was done on the day of atonement, during that time, from when David became king, until Solomon had finished building the temple?

Did the high priest bring the blood from the tabernacle in Gibeon, to the ark in the city of David?

I emailed  Ask the Rabbi - Jewish Questions, Jewish Advice - Ask A Rabbi Now! (chabad.org) to try and get an answer to your question by asking the questions below:

"When the tabernacle of the congregation was at Gibeon (2 Chronicles 1:3) and the ark of the covenant was inside a tent in Jerusalem (2 Chronicles 1:4) while the first temple was being built by King Solomon, what happened on Yom Kippur, the Day of Atonement, since the mercy seat on the ark of covenant that is sprinkled with blood on Yom Kippur was not inside the Holy of Holies inside the tabernacle of the congregation at Gibeon?  Was Yom Kippur cancelled, or did the high priest take the blood from the animal that was sacrificed inside the tabernacle of the congregation at Gibeon and travel with it  to sprinkle the mercy seat on the ark of the covenant which was inside a tent in Jerusalem?"

Below is the Rabbi's reply indicating that he does know the answer to my questions, but if he ever finds out he will let me know the answer:

Hi Donna,

"Thank you for your inquiry and please accept my apologies for the delayed response.

Great question.

These verses indicate that there was a sanctuary, albeit with a lesser degree of sanctity.

Offerings that had fixed times were allowed and, indeed, brought to the sanctuary without the ark.

Offerings that their times were not fixed were not allowed to be brought to the Mishkan in Gibeon.

In later times during the second temple, where the ark was not present in the holy of holies as King Yoshiyahu hid it, they did do the service of Yom Kippur. Yet to infer that this was also the case in Gibeon is likely, as mentioned, all the sacrifices that had fixed times were offered in the Tabernacle while in Gibeon.

I have yet to see your question addressed directly, on how the blood was sprinkled on Yom Kippur in this situation. With G-d's help, a resolution will be found, and when I do find one, I will share it with you."

Best regards,

Rabbi Elchonon Kazen