The Pasuk writes ‘Chag HaSuccot Taase Licha.’ From this Pasuk, there are some Rishonim, Rashi specifically, and some Geonim like Rav Chai Gaon, who say there is a Mitzvah itself to build the succah. Which means besides the Mitzvah of sitting in the succah and sleeping in the succah, there is a separate Mitzvah to actually build the succah. However, there are those that argue on this. Among them, is the opinion of the Rasbash (Rav Shlomo Duran), who says that building the succah is only in preparation for the Mitzvah which is sitting and dwelling in the succah.
We follow that second opinion, that although it’s a Mitzvah to build the succah, it’s more of a preparatory Mitzvah to lead to the main Mitzvah of dwelling in the succah. That is why we do not make a Beracha when we are building the succah. We don’t say ‘Asher Kidishanu BiMitzvotav VeTzivanu Livnot succah’.
That being said, the Rama brings down at the end of the laws of Yom Kippur, that after Kippur is over one should right away begin building the succah. And the reason is based on a pasuk that says ‘Yelchu MeChayil El Chayil,’ meaning we should go from strength to strength. We just finished our Teshuva process, and we want to show that that process was effective, so right way we want to start the new slate involving ourselves in Mitzvah. It should be pointed out that if someone, for whatever reason, can not actually start the building the succah, so the Kaf Hachayim brings down, that he should at least start discussing the plans of the building of the succah. Some people go to the spot and start mapping it out, or they begin with putting up one wall. It is said that the Gaon MeVilna used to start learning Masechet succah on Motzae Kippur. His form of immediate preparation after Kippur was to learn the Halachot. Therefore, one should try to do something succah related on Motzae Kippur in order to go from one strength to another strength.
Since there is a preparatory Mitzvah in building the succah, one should try to involve himself in some part of the construction and not totally delegate it to a messenger. We have a principal that says it is always better to do the Mitzvot yourself rather than delegate. One should at least have some involvement in putting up the succah. Specifically, one should try and put up the Sechach (roofing), which is the main part of the succah. Even though you might have someone put up the frames or walls, you should at least be involved in putting up some or all of the Sechach. This is also brought down from the Kaf Hachayim.
So to review, there is a Machloket if building the succah is a Mitzvah itself. We hold that it is a preparatory Mitzvah. There is no Beracha in the building, but one should involve himself in the construction. Specifically, try to begin on Motzae Yom Kippur to build the succah or start learning the Halachot in regards to the succah.