Mishna in Makoth 2:7
וְכֹהֵן גָּדוֹל שֶׁהָרַג , אֵינוֹ יוֹצֵא מִשָּׁם לְעוֹלָם
A Kohain Gadol who kills goes to the Ir Miklat forever.
They would have to appoint another one, to do the Avoda on Yom Kippur, as the first one loses his job as Kohain Gadol, as the Rambam הלכות רוצח ושמירת נפש at (7:14) says about all those sentenced to Ir Miklat, after they can go home:
אַף עַל פִּי שֶׁנִּתְכַּפֵּר לוֹ אֵינוֹ חוֹזֵר לַשְּׂרָרָה שֶׁהָיָה בָּהּ לְעוֹלָם. אֶלָּא הֲרֵי הוּא מוּרָד מִגְּדֻלָּתוֹ כָּל יָמָיו הוֹאִיל וּבָאָה תַּקָּלָה זוֹ הַגְּדוֹלָה עַל יָדוֹ:
The Rambam in הלכות רוצח ושמירת נפש at (7:10) actually says that this "eternal sentence" is only if there was no another Kohain Gadol appointed at the same time.
וְכֹהֵן גָּדוֹל שֶׁהָרַג וְלֹא הָיָה שָׁם כֹּהֵן גָּדוֹל אַחֵר. הֲרֵי אֵלּוּ גּוֹלִין וְאֵינָן יוֹצְאִין מֵעִיר מִקְלָט לְעוֹלָם
If the Kohain Gadol co-existed with another Kohain Gadol (e.g. one of them became ineligible to perform the Yom Kippur service because he became tamei (ritually impure) so the "back up" or Assistant Kohain Gadol took his place) then the accidental killer would be permitted to leave the ir miklat, the city of refuge, when the other one died. But he would not get his position back, as above.