Q. Since Tisha Beav is called Moed or a Yom Tov, can an Avel siting Shiva go to shul and daven at the amud?
A. Shulchan Aruch (O.H. 559: 6) rules that an Avel attends shul on Tisha Beav in the night and also in the day until the end of the Kinos recitation.
Mishna Berura (24) explains that although usually Avelim do not exit their home during the first three days, in this case, since we are all in mourning, he may do so. He quotes opinions that differ and maintain that on the first three days he should not attend. He further quotes a leveling position from R”Sh Kluger zt’l that at night since there are fewer Kinos he should better remain home. However in the day, when there are many Kinnos and he would likely be pained for missing them he should attend. After three days, he may attend at night too.
Nitei Gavriel (Avelus 125: 1) mentions quoting Magen Avrohom, that If he is by himself, he may attend at night also. However, he should refrain attending for Mincha.
Sidur Yaavetz maintains an Avel should not daven for the Amud on Tisha Beav. Similarly Mikroei Kodesh (8: 52) quotes some opinions that maintain a mourner should indeed not direct the services during Tisha Beav as it also called a Moed. However, he mentions that most Poskim are lenient. Adonei Paz and Nitei Gavriel (Ben Hametzarim 1: 58: 10) rule likewise.
Horav Shlomo Miller’s Shlit’a opinion is similar.
Rabbi A. Bartfeld as advised by Horav Shlomo Miller and Horav Aharon Miller Shlit’a