Is there laundry that is permissible to wash during the 9 days? Kids' clothes, undergarments, socks, etc.? I've heard yes, but I'm looking for a source. (This blog post cites the RaM"A but doesn't fully explain it.)
Per Rabbi Eli Mansour's halakhic online responsa:
As was mentioned, it is forbidden to wear freshly laundered garments the week of Tisha BeAv, and it is forbidden to launder garments during the week of Tisha BeAv. … The only exception is, if it is for infants and toddlers up to 3 years old. It would be permissible since their garments become soiled very easily. It is also permissible to wash their cloth diapers, (in the event one still uses cloth diapers). But aside for little children, it is forbidden to wash clothes. It is forbidden not only by a Jew, but even by a non-Jew.
See below for links to more of Rabbi Mansour's halakhic opinions on this topic.
Unfortunately, I once had to ask a shiala because my daughter had been discovered to have lice on the afternoon before Tisha b'Av, and the protocol is to launder all the clothes that might have come into contact and have lice or lice eggs on them. My own rabbi was out of town, so I called Rabbi Barry Fruendel of Kesher Israel Congregation in Washington, D.C. He said that not only could I launder all of the clothes in our house on Tisha b'Av, I had to do the laundry because it was a health issue. We did 17 loads (my wife deemed everything in the closets at risk). Thankfully, her camps thereafter did lice checks well before the 9 days, and she never got lice again.
A secondary source would be the English Sefer "Summary of Halachos of the Three Weeks" by Rabbi Shimon Eider zt"l, chapter III D on page 9.
You can see it here with all the sources: http://books.google.co.il/books?id=k2lV5wQwwj0C&lpg=PP1&hl=iw&pg=PA9#v=onepage&q&f=false
Here is a copy of the relevant chapters - without the sources (which are in Hebrew and time consuming to type).
HALACHOS OF THE THREE WEEKS - Ch III D
3. A Jew cleaning clothing of a gentile
A Jew may clean or launder garments of a gentile during the Nine Days-but should refrain from cleaning their garments in public during the week Tisha B'Av occurs.
4. Only one garment
If a person has only one garment and it requires cleaning or laundering, or if he has many but all are dirty, it is permissible to clean a garment until the Shabbos before Tisha B'Av, but cleaning is prohibited during the week in which Tisha B'Av occurs.
6. Washing children's clothing
Washing children's diapers and clothing are permissible even in the week in which Tisha B'Av occurs. This halacha applies even to children whop do not wear diapers but who constantly soil their clothing with dirt, mud and the like. Washing these garments, however, should be done in private.
7. Washing for the purpose of a mitzvah
Washing for the purpose of a mitzvah is permissible-even during the week in which Tisha B'Av occurs. Therefore, a woman who is required to wear white for the Seven Clean Days may wash her garments and sheets for this purpose. On Tisha B'Av, however, if she discovers that she has no clean white garments, washing her garments is prohibited; she should wear a clean (i.e. without stains)-even colored garment-instead.