The Rama (Rav Moshe Isserles of Cracow, 1530-1572) writes (in Orah Haim 124) that the custom among Ashkenazic communities is to stand during the Hazzan’s repetition of the Amida. The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) cites the Peri Megadim (Rav Yosef Teomim, 1727-1792) as bemoaning the fact that many people do not follow this practice, as some stand while others sit.
An earlier source for this practice is the Rambam (Rav Moshe Maimonides, Spain-Egypt, 1135-1204), who, in Hilchot Tefila, writes that during the Hazzan’s repetition, "Kol Ha’am Omedim Ve’shom’im" – "all the people stand and listen." The Rambam clearly writes that the congregation should stand during the repetition of the Amida.
This is, indeed, the view of numerous Sephardic Poskim, including the Hid"a (Rav Haim Yosef David Azulai, 1724-1806), in Kesher Godal; Rav Haim Palachi (Turkey, 1788-1868); Rav Yaakob Haim Sofer (Baghdad-Jerusalem, 1870-1939); the Hesed La’alafim (Rav Eliezer Papo, 1785-1828); and the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Parashat Teruma.
By contrast, Hacham Ovadia Hedaya (Egypt-Israel, 1889-1969), in his work Yaskil Abdi, contended that the Rambam’s comments must be read differently. The Gemara in Masechet Sota establishes the prohibition against speaking during Torah reading from the verse in the Book of Nehemya (8) which describes what happened when the Sefer Torah was open: "U’be’pit’ho Amedu Ha’am" – "When it was opened, the nation stood." This verse indicated to the Gemara that the congregation must remain silent once the Torah scroll is opened to be read. Clearly, the Gemara understood that the word "Amedu" means not "stand," but rather "remain silent." By extension, Hacham Ovadia Hedaya writes, when the Rambam says that people must be "Omedim Ve’shom’im" during the repetition of the Amida, he means not that they should stand, but rather that they must remain silent.
Hacham Ovadia Hedaya draws proof to his theory from the fact that the Rama does not cite the Rambam as his source for the practice to stand. He writes that this was customary in Ashkenaz, without saying that this was the Rambam’s view. Evidently, the Rama did not understand the Rambam to mean that the congregation must stand during the Hazzan’s repetition of the Amida. If he did, he would have drawn our attention to the fact that the Rambam is the source of this Ashkenazic practice.
Hacham Ovadia Hedaya adds that in any event, nowadays, when we all recite the Amida ourselves, and nobody fulfills their requirement by listening to the Hazzan’s repetition (as people did in yesteryear), there is no reason for the congregation to stand. To the contrary, it might be preferable to sit, to make it easier to concentrate on the Hazzan.
One can easily refute Hacham Ovadia Hedaya’s arguments, in light of the fact that the Rambam, in establishing the requirement to remain silent during the Torah reading, writes, "Kol Ha’am Shotekim Ve’shom’im" – "all the people must be silent and listen." In the context of the Torah reading, the Rambam uses the word "Shotekim" to refer to silence, and thus it stands to reason that the word "Omedim" which he uses in reference to the Hazzan’s repetition means "stand." Otherwise, he would have used the same term in both contexts. Moreover, the Rambam’s son, Rabbenu Abraham Ben Ha’Rambam (1186-1237), cited in the introduction to the work Ma’aseh Roke’ah, tells of how the Rambam did away with the Hazzan’s repetition of the Amida in his synagogue. Seeing that people were not paying attention to the repetition, and were conversing with one another, the Rambam decided to eliminate the repetition. In describing the situation which led the Rambam to this drastic decision, Rabbenu Abraham writes that the people "were not standing with awe and fear like they stand during the silent Amida," as is required. This clearly indicates that the Rambam felt that standing is required during the repetition.
As for the Rama, he was just recording the practice among Ashkenazic communities. His purpose was not to document the Halachic sources of this practice, but simply to establish that this was the accepted custom in his region.
Some have suggested drawing proof that standing is not required from the comment in the Talmud Yerushalmi (Masechet Rosh Hashanah) that Rav Hisda would sit during the repetition of the Amida of Musaf on Rosh Hashanah. This would appear to prove that it is acceptable to sit during the repetition of the Amida. However, the Halachot Ketanot (Rav Yaakob Hagiz, 1620-1674) refutes this proof, noting that Rav Hisda likely sat because he was elderly and frail, and it is clear that somebody in this condition is allowed to sit. Indeed, even the Ben Ish Hai, who, as mentioned, requires standing during the repetition, concedes that an elderly or otherwise frail person may sit.
Regardless, Hacham Ovadia Yosef, in Yehaveh Da’at, brings proof against the stringent view, showing that it is legitimate to sit during the repetition of the Amida. His son, Hacham David Yosef, notes that the common practice among Sephardim nowadays is to sit, though he shows that this was not always the case.
In practice, those who have the custom to sit certainly have a basis on which to rely, but one who is healthy and can easily stand should preferably do so, in accordance with the view of the Poskim noted above.
Summary: The general custom among Sephardim is to sit during the Hazzan’s repetition of the Amida. Although this practice is valid, it is preferable for those who are healthy and do not have a problem standing to do so.