When the first day of Sukkot falls on Shabbat, we do not take the Lulab and Etrog on this day. The Sages suspended the Misva of Arba Minim (four species) in such a case out of the concern that somebody might forgetfully carry the Lulab and Etrog in a public domain, in violation of Shabbat. This is the same reason why we do not sound the Shofar on the first day of Rosh Hashanah when it falls on Shabbat.
It thus emerges that in a year when the first day of Sukkot falls on Shabbat, we do not observe the Torah obligation of Arba Minim. The Torah commands taking the four species on the first day (Vayikra 23:40); throughout the rest of Sukkot, we take the Arba Minim by force of Rabbinic enactment. It is only in the Bet Ha’mikdash where the Torah requires taking Arba Minim all seven days of Sukkot. Thus, if the first day falls on Shabbat, in which case we do not take the Lulab and Etrog on that day, we fulfill only the Rabbinic obligation of Arba Minim, but not the Biblical command.
In this case, we recite the Beracha of "Shehehiyanu" when we take the Lulab and Etrog for the first time on Sunday. Since we fulfill the Rabbinic obligation for the first time that year, we recite "Shehehiyanu," even though we do not fulfill the Torah obligation.
If a person realizes on Yom Tob that he does not have Arabot, may he ask a gentile to pick Arabot for him?
The Rashba (Rabbi Shlomo Ben Aderet of Barcelona, 1235-1310) addresses this question and rules that one may not ask a gentile to pick Arabot for him in such a case. Even though asking a gentile to perform Melacha (forbidden activity) on Yom Tob is forbidden only by force of Rabbinic enactment, one may not make such a request even for the purpose of a Misva. The Arabot picked by a gentile on Yom Tob have the status of Mukse, since they were attached to the tree and hence "off limits" when Yom Tob began, and they therefore may not be used even if a gentile picks them. This is the ruling codified by Hacham Ovadia Yosef, in his work Hazon Ovadia (listen to audio recording for precise citation).
An interesting variation of this question arises in a situation of a Jew who lives in the Diaspora and visits Israel for Sukkot. As a Diaspora resident, he must observe two days of Yom Tob, even though residents of Israel observe only a single day. If the visitor realizes on the second day of Sukkot that he does not have Arabot for the Misva, may he ask an Israeli to pick Arabot for him, given that it is permissible for Israelis to do Melacha on the second day of Sukkot?
Rav Shlomo Zalman Auerbach (Israel, 1910-1995) addresses this question in his work Minhat Shelomo (vol. 2, Siman 45), and rules that a visitor from the Diaspora indeed may ask an Israeli to pick Arabot for him in this case. This situation differs from the case of asking a gentile, in that this day is not Yom Tob at all as far as the Israeli is concerned. Therefore, even though one may not ask a gentile to pick Arabot for him on Yom Tob, someone visiting Israel from the Diaspora may ask an Israeli resident to pick Arabot for him on the second day of Sukkot.
Summary: When the first day of Sukkot falls on Shabbat, we do not take the Arba Minim, and we recite "Shehehiyanu" when we take the Arba Minim on the second day. A person who realizes on Yom Tob that he does not have Arabot may not ask a gentile to pick Arabot for him. However, someone visiting Israel from the Diaspora may ask an Israeli resident to pick Arabot for him on the second day of Sukkot, since Israeli residents do not observe that day as Yom Tob.
