The Rosh (Yoma 8:23), the Mordechai (at the beginning of his notes to Masechet Yoma), and the Tur (Orach Chaim 605) record this practice (mentioned in the aforementioned Rashi on Shabbos 81b) with approval. They, however, mention that the usual practice is to take a chicken and slaughter it. They also note that the ritual is performed on Erev Yom Kippur. The Rosh explains that the Gemara sometimes refers to a chicken as a Gever (see Yoma 20b), which also means man. Thus, a chicken is an appropriate substitute for man. He also offers a pragmatic explanation: that chickens are readily available and less expensive than larger animals such as a ram.
The Rashba does acknowledge that all of the Ashkenazic rabbis of his time practiced Kapparos and that the practice is recorded in the writings of Rav Hai Gaon. The Rama notes that this practice is recorded as early as the Geonic period and is the accepted practice in all Ashkenazic communities. The Rama regards the practice as a Minhag Vasikin, a venerated practice that one must not neglect. The practice recorded in the Rama is to slaughter a chicken for every family member.
The Ben Ish Chai (Parshas Vayelech 2), Kaf Hachaim (605:8), and Rav Ovadia Yosef (Teshuvot Yechaveh Daat 2:71) record that Sephardic Jews have adopted this custom despite the opposition of Rav Yosef Karo, the author of the Shulchan Aruch. An explanation for this change is that the Ari zt”l enthusiastically embraced this practice (as noted by the Magen Avraham 605:1) based on his Kabbalistic approach. The Ari zt”l has an enormous impact on Sephardic practice in a wide variety of areas.
The Mishna Berura cites the Pri Megadim who rules that Kapparos may be performed throughout the Aseres Yemai Teshuva. Indeed, Rashi records that this ritual is performed on Erev Rosh Hashanah. Accordingly, the Mishna Berura suggests that Kapparot be performed a day or two before Erev Yom Kippur to relieve the stress on the Shochtim. Rav Ovadia Yosef writes that Kapparos may be performed throughout the entire Aseres Yemai Teshuva.