Gemara

Beraisa: Anyone obligated to immerse may do so normally on Yom Kipur or Tish'ah b'Av;

R. Chanina Segan ha'Kohanim says, the Beis ha'Mikdash is worthy that (amidst mourning its loss) we forfeit immersing once a year (on Tish'ah b'Av).

Yoma 88a - Beraisa #1: Anyone obligated to immerse may do so normally on Yom Kipur:

A Nidah or Yoledes may immerse on Yom Kipur night;

A Ba'al Keri may immerse at any time until Minchah.

R. Yosi permits at any time on Yom Kipur.

Contradiction (Beraisa #2): The following may immerse normally on Yom Kipur - a Zav, Zavah, Metzora, Metzora'as, Bo'el Nidah, and Tamei Mes;

A Nidah or Yoledes may immerse on Yom Kipur night;

Ba'al Keri may immerse at any time on Yom Kipur;

R. Yosi says, he may (some texts - need) not immerse after Minchah.

Answer: If the Ba'al Keri already prayed Ne'ilah, R. Yosi permits until Minchah; if not, he permits the entire day.

Suggestion: Chachamim hold that Tevilah bi'Zmanah Mitzvah, and R. Yosi holds that Tevilah bi'Zmanah Lav Mitzvah.

Question - Beraisa #3: If Hash-m's name was written on a person...and he has a Mitzvah to immerse, he wraps a reed around the name;

R. Yosi says, he immerses normally.

The first Tana holds that Tevilah bi'Zmanah Lav Mitzvah, and R. Yosi holds that it is a Mitzvah!

Rather, Beraisa #2 should say that R. Yosi b'Rebbi Yehudah permits only until Minchah. (Regarding a woman (Nidah 29b) who does not know when or what she miscarried, and saw blood for three straight weeks...others say that she is Tovel 95 times or 35 times.) He says that it suffices for her to immerse once at the end.

Rishonim

Rambam (Hilchos Shevisas Asor 3:2): All Chayavei Tevilos immerse normally on Yom Kipur or Tish'ah b'Av.

Magid Mishnah: This is like the first Tana (Ta'anis 13a).

Hagahos Maimoniyos (5): Tana'im argue only about Tish'ah b'Av, but all permit immersing on Yom Kipur. The Ri rules like R. Chanina Segan ha'Kohanim. Perhaps the Rambam does also. In Hilchos Tish'ah b'Av he does not mention immersing! Perhaps the words 'or Tish'ah b'Av' here are a scribal error, i.e. the scribe adopted the text of the Beraisa.

Rambam (3): Nowadays, if someone saw Keri on Yom Kipur he cleans the area but may not immerse, for today one who immerses is still Tamei Mes. Immersion is not required to pray, it is only a custom. A custom may forbid what is permitted; it cannot permit what is forbidden (bathing on Yom Kipur).

Bach (OC Sof Siman 613): The Rambam forbids a Ba'al Keri even though he holds that Tevilah bi'Zmanah Mitzvah and permits other Chayavei Tevilos.

Note: The Rambam alludes to the reason; a Ba'al Keri who immerses today is still Tamei Mes, so the Tevilah did not help. Nowadays Tevilah helps only for women, to permit them to their husbands.

Rif (Sof Yoma) and Rosh (8:21): If one sees Keri on Yom Kipur he immerses; if it is still moist he need not rub his skin to clean it...

Ran: This was only when it was forbidden for a Ba'al Keri to pray; nowadays, a Ba'al Keri may pray, so he may not immerse, like the Rambam says.

Migdal Oz (Hilchos Shevisas Asor 3:3): The Rif permits a Ba'al Keri to pray, so surely he forbids immersing nowadays, like the Rambam.

Note: The Magid Mishnah (ibid.) and Ran seem to agree. They must explain that the Rif (who brings only Halachos that apply nowadays) brings the Beraisa verbatim merely to teach about cleaning the area. (See the Bach below.) However, the Beis Yosef (OC 613 DH ul'Inyan Tevilas) connotes that the Rif and Rosh hold like the Ri Bartzeloni and BaHaG (cited in Tur 613) who permit immersing nowadays. However, the Shulchan Aruch, which is based on the majority opinion of the Rif, Rambam and Rosh, forbids like the Rambam! Perhaps the Beis Yosef rules like the Rambam because the Rif and Rosh are not explicit. They seem to permit Tevilah nowadays, but they could agree with the Rambam's compelling reasoning. Similarly, the Tur usually follows (and never ignores) the opinion of his father, yet here he does not mention the Rosh (or Rif).

Rashi (Shabbos 121a DH Tovel): Immersion of Reshus is forbidden on Yom Kipur. The opinion that permits the entire day holds that Tevilah bi'Zmanah Mitzvah; the opinion that permits until Minchah permits for the sake of Tefilah.

Tosfos (ibid. DH Hachi): The text should say 'R. Yosi does not require immersing after Minchah.' The first Tana says that a Ba'al Keri immerses until Minchah, i.e. he holds that Tevilah bi'Zmanah Lav Mitzvah. Nevertheless, he permits all Chayavei Tevilos to immerse on Yom Kipur, even a Tamei Mes and Tamei Sheretz who need not immerse to pray.

Rosh (Ta'anis 4:39): One may wash before Tefilah on Tish'ah b'Av, just like Chayavei Tevilos may immerse on Yom Kipur and Tish'ah b'Av. Only bathing for pleasure is forbidden.

Bach (554 DH u'Mah she'Chasav u'Mihu): Perhaps the Rosh and Tur merely teach why one may wash for Tefilah, but they do not actually permit Tevilah.

Poskim

Shulchan Aruch (OC 554:8): Tevilah bi'Zmanah is permitted on Tish'ah b'Av. However, nowadays women do not immerse bi'Zman, so it is forbidden.

Beis Yosef (DH Aval ha'Tosfos, citing Tosfos Beitzah 18b DH Kol): Even though Tevilah bi'Zmanah is permitted on Yom Kipur and Tish'ah b'Av, this does not apply nowadays, for women do not immerse bi'Zmanan mid'Oraisa (they count seven clean days after any sighting of blood). However, a Tana permits a Ba'al Keri to immerse until Minchah. Even though he holds that Tevilah bi'Zmanah Lav Mitzvah (otherwise, he would permit the entire day), he allows immersing to pray in Taharah. We do not immerse on Tish'ah b'Av, like R. Chanina Segan ha'Kohanim; the Yerushalmi says that the Halachah follows him. The Ri says that nowadays we do not immerse even on Yom Kipur; it was permitted only when there was concern not to Metamei Taharos. Nowadays women immerse only to permit relations. Since this is forbidden on Yom Kipur and Tish'ah b'Av, the immersion is postponed.

Beis Yosef (DH v'Rabeinu): Nowadays women do not immerse even during Aveilus, all the more so not on Yom Kipur or Tish'ah b'Av!

Question: Tosfos (Shabbos 121 DH Hachi) proved that Chayavei Tevilos may immerse on Yom Kipur, even according to the opinion that Tevilah bi'Zmanah Lav Mitzvah!

Answer #1 (Tosfos Yoma 88a DH k'Savrei): No - the Tana merely taught that the need to immerse for the sake of Tefilah is only until Minchah, because Ne'ilah may be said at night, but in any case one must immerse because Tevilah bi'Zmanah Mitzvah.

Answer #2 (Gra DH Hilkach, citing Ramban): The Tana teaches that one who was Ba'al Keri from before Yom Kipur may immerse only until Minchah, but one who saw Keri on Yom Kipur could immerse the entire day because Tevilah bi'Zmanah Mitzvah.

Shulchan Aruch (613:11): Nowadays, if someone saw Keri on Yom Kipur he cleans the area and prays. He may not immerse, even if he normally immerses for prayer.

Bach (DH ul'Inyan Tevilas): The Ri Bartzeloni permits one who saw Keri on Yom Kipur to immerse for the sake of Tefilah, not because Tevilah bi'Zmanah Mitzvah. The Beraisa discusses seeing Keri on Yom Kipur to teach about cleaning moist and dried semen. He did not bother to permit even one who was Ba'al Keri from before Yom Kipur, for everyone normally immerses on Erev Yom Kipur.

Shulchan Aruch (ibid. 12): Nowadays, a woman may not immerse on Yom Kipur, even if it is her first opportunity.