You are supposed to be tahor before Yom Tov for the taharos of the Yom Tov.

Bais Shammai says, if Yom Tov comes out right after Shabbos then utensils and the person themselves should go the mikvah before Shabbos. Bais Hillel say the utensils must be done before Shabbos but the person himself can go to the mikvah on Shabbos.

Beitzah 2:3

Hashakah is when you take tamai water and combine it with mikvah water making it tahor.

Bais Shammai & Bais Hillel both agree that hashaka can be done on Yom Tov in a stone utensil that cannot become tamai.

Chazal said it is permitted to bring a utensil to the mikvah, not because it is tamai but rather to make it holier, such as utensils used for chulin being brought to the mikvah in order to use them for terumah. Since they are not actually tamai, they may be brought to the mikvah on Yom Tov.