As we explained, if a person didn't bring a korban on the first day, he can bring the korban on all seven or eight days of the yom tov. The Gemara (Chagigah 9a) states a machlokes whether the korbanos brought are to make up for the korbanos that were missed on the first day (דראשון תשלומין), or it is to fulfill a new obligation he has of bringing korbanos on the day he arrives at the Beis Hamikdash, which was during chol hamoed (לזה זה תשלומין). According to this view, the korbanos aren't making up for a previous debt; rather, it is a new debt. You came to the Beis HaMikdash today, and you must bring korbanos today.
Tosfos (לזה זה תשלומין ה"ד) writes that all opinions will agree that Shavuos is certainly for the first day, דראשון תשלומין. If he didn't bring the korbanos on the first day and brings them during the week, he is fulfilling the obligation that he missed on the first day. Tosfos explains that we can't say he is bringing korbanos for the day he came to the Beis HaMikdash (לזה זה תשלומין) because it is a weekday. Therefore, it is more logical to say that the korbanos are to complete the obligation of the first day (דראשון תשלומין).
So, according to Tosfos, during the week, we make up for what we missed on Shavuos. This is another indication that the entire week is still Shavuos, and we can make up for what we missed during Shavuos in this week.
The Tchortkover zt'l and other tzaddikim would say, שבועות נאך איז שבועות נאך שבת, "Shabbos after Shavuos is still Shavuos." (The word נאך means "after" and it means "still", depending on the context. So Shabbos after Shavuos שבועות נאך is שבועות נאך, still Shavuos.)
Chazal (Yoma 21b) say, רגלים לעולי אותו שמגביהין כסדורו שסילוקו המקום לפני חיבתכם ראו להם ואומרים, "The [kohanim] would lift the Shulchan to show it to the people who came up to the Beis HaMikdash for yom tov. They would say to them, 'See how much Hashem loves you! The breads are taken off the Shulchan [as hot as] on the day they were placed on the Shulchan.'"
The Berzhaner Rav zt'l points out that one of the three yomim tovim when there is a mitzvah of לרגל עליה is Shavuos, and it seems that the Gemara is referring to this yom tov as well. Let's say Shavuos was on Monday or Tuesday, how did the kohanim show the nation the hot הפנים לחם on Shabbos? This is proof that they remained in the Beis HaMikdash for a week after Shavuos. Why did they stay? Because the entire week is still somewhat Shavuos. And therefore, if someone wasn't ready to receive the Torah on Shavuos, he can do so this week.
The Beis Avraham (Naso) quotes the Be'er Mayim Chaim (Bishalach) as stating that Dasan and Aviram weren't at the main kriyas Yam Suf. When they heard that the sea split for the Jewish nation, they went to the Yam Suf themselves, and the sea split just for them. Similarly, the Beis Avraham said, Hashem gave the Torah on Shavuos. Those who weren't "present" (with their hearts) to receive the Torah can still receive the Torah now. On this Shabbos, Hashem gives them another chance.
Kabalos Tovos
One way to attach oneself to the yom tov is to take on a kabbalah tovah, even if it is small, and that will be the rope that ties him to the yom tov, even after it passes.
