B"H
Every chag has tashlumin, "make up time", to bring the required offerings on it, all 7 days.
Even though shavuos is only 1 day, for 6 days after it, in the times of the Beis Hamikdash, people would be able to bring the necessary sacrifices.
Nowadays, many Jewish communities that I'm aware of do not say tachnun for 7 days beginning with shavuos. Why? We don't offer sacrifices anymore.
On the chag itself I can understand why we don't say tachnun, since we make kiddish, refrain from melacha, and the entire day is "holy".
However, during the make-up time days, we do so melacha, we don't make kiddish, and they are in general just regular days.
This is also unlike the tashlumin for other holidays, which we don't say tachanun on, because that are actually a part of the festival itself (chol hamoed).
So why skip say tachnun on the shavuos tashlumin days, which are seemingly just regular days (only that if there was a beis hamikdash we would be able to offer the necessary sacrifices)?
Similar question may apply to pesach sheini, although that holiday has it's own actions that we do on it even nowadays (eating matzah), so it would make more sense why we treat it as a holiday now and don't say tachnun, but I'm not aware of any unique action that people are supposed to do on the tashlumin days of shavuos that would warrant treating them as their own holiday to not say tachnun