The seven clean days need to be completed before immersion. They cannot be shortened, even if a couple will refrain from relations.
In cases such as these, it may be possible to shorten the five-day wait prior to beginning the clean days. If by some chance you performed a bedikah on Shabbat, please get back to us with details.
Sometimes, walking to the mikveh and then back would be too far, but just walking home would be possible. If there is a possibility that you could be dropped off at the mikveh before yom tov and manage the one-way walk back, that would be worth exploring. You can accept yom tov after you get out of the car, and light candles from an existing flame when you get home.
If there is absolutely no way to immerse on Sunday night, then immersion will have to wait until after yom tov. Most preparations can be performed on Sunday, but you should shower, comb your hair, brush your teeth, and inspect yourself again just prior to immersing.
Chag Same’ach, and please feel free to get back to us with any further questions.