Mazal tov on the birth of your baby!
Mikveh immersion after birth is permissible on Shabbat or Yom Tov, or Motzei Shabbat or Motzei Yom Tov. So if mikveh night falls out on Shavuot night you can immerse then.
It’s true that there are some halachic considerations that make immersion on Shabbat or Yom Tov more complicated (such as immersing the body in heated water). However, there are also halachic considerations (such as allowing intimacy) that encourage immersion at the earliest possible time, and these considerations take precedence. Therefore, mikveh immersion is permitted as soon as you have completed the seven clean days, even if it falls out on Shabbat or Yom Tov. This is true whether the immersion follows a regular period or is after childbirth.
More information about immersion after childbirth can be found here, and more information about tevilah on Shabbat or Yom Tov can be found here.
We hope this helps. Please write back with any further questions. Chag same’ach!