May a mourner lead the prayers during Chanukah?[1]

A mourner who is within 30 days of mourning for a relative, or within 12 months of mourning for a parent, may lead the services on Chanukah, with exception to the reading of Hallel.[2] [Thus, after Chazaras Hashatz, another individual is to switch with him and lead the Hallel prayer. The mourner is to switch back immediately after the completion of Hallel, and he is to recite the Kaddish that is said after Hallel.[3]] On Rosh Chodesh, however, and likewise on any day on which Mussaf is said, he does not lead the services, even during Minchah or Maariv.[4]

May one who has a Yartzite on Chanukah lead the Hallel?

Yes.[5]

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[1] Admur 131:5; Maharil 22; Elya Raba 582:22; Gilyon Maharsha 376:4 in name of Kneses Hagedola 391; M”B 581:7 in name of Machatzis Hashekel; Gesher Hachaim 23:5; Sefer Haminhagim p. 161 [English]; Igros Kodesh 2:324; 8:236; 11:3; 24:149 [published in Shulchan Menachem 5:303-304]; See Maharam Shick 183; Poskim in Nitei Gavriel 58:1; Piskeiy Teshuvos 683:2

Other opinions: Some Poskim rule a mourner is not to lead the prayers of Shacharis during Chanukah, but may lead the prayers of Mincha and Maariv. [P”M 683 M”Z 1; M”B 683:1; 671:44 based on P”M; 581:7 in name of Gr”a; Kaf Hachaim 683:5; Maharam Shick 183 in name of Chasam Sofer; Aruch Hashulchan Y.D. 376:14; Dvar Moshe 84 that so is custom of Chassidim;] Some are accustomed that mourners do not Daven for the Amud at all on Chanukah. [See Divrei Yatziv Y.D 236 based on Ramak; Implication of Shut Maharil 22; Mahram Mintz 43; Noam Megadim p. 43; Yad Sofer 37 in name of Chasam Sofer; M”B 132 “Kuntrus Mamar Kaddeishim”; Piskeiy Teshuvos ibid that so is custom in majority of Chassidic communities; Rivivos Efraim 1:443 in name of Rav Moshe Feinstein; See Piskeiy Teshuvos 132:31]

[2] The reason: As the congregation is now found in a state of joy which is similar to Shabbos and Yom Tov. [Admur ibid; Maharil ibid; Elya Raba ibid]

[3] Hiskashrus 908 p. 15

[4] Sefer Haminhagim ibid

[5] The reason: As the above law only applies to an Avel who is in a state of mourning and is thus not to lead the prayers on Shabbos and Yom Tov. This does not apply by one who has a Yartzite.