The mishna in Taanis 26a says that it is forbidden to do laundry during the week in which Tisha B’av falls out. The Shulchan Aruch 551:3 brings this as the halacha. The Rama adds that we have a custom not to do laundry starting from Rosh Chodesh, including Rosh Chodesh. Being that from Rosh Chodesh until Shabbos is only a custom, some leniencies apply. Mishna Berura 551:29 says that someone who ran out of clean cloths is permitted before Shabbos to wash something that is needed to be worn through Tisha B’av. After Shabbos until Tisha B’av this is forbidden.
The prohibition to do laundry also applies to a non-jew or a machine to wash the clothing for you. It also applies even if you are not planning on wearing the washed cloths until after Tisha B’av.
Regarding whether or not it is permitted to wash children’s cloths is subject to controversy. Most sources only allow to wash cloths of a child until the age of three, although some permit to wash the cloths of all children. From Rosh Chodesh until Shabbos is even more lenient. The reason is because the mitzvah of mourning for a child is chinuch, to train them in mitzvos. For chinuch it is enough to do what is forbidden in the mishna from after Shabbos until Tisha B’av, and not from Rosh Chodesh. Therefore, if one needs to launder children’s clothing, it is best to do so before Shabbos. See Chayai Adam 133:18.

Rabbi Tzvi Aryeh HymanMir Yeshiva, Jerusalem- [email protected]
Shabbat Shalom