Sources of pure soap laundry powder or liquid
Always check ingredients first, manufacturers change product ingredients all the time.
A small % of Amazon purchases linked through here goes to SolveEczema if purchased through the link, on the order of $15/month).
To save money or time, do it yourself! There are many online recipes for laundry powder and gel using bar soaps, and various amounts of borax, washing soda, and baking soda.
Here are the pre-packaged products I found:

Laundry powder, soap flakes, soap bars
Pure Soap Flake Company
http://puresoapflakes.com/
They have an unscented laundry powder which is just simple soap, baking soda, borax and washing soda.
************************************************************************
Cal Ben Seafoam Laundry Soap
http://www.calbenpuresoap.com/seafoam-laundry-soap.php
More complicated list of ingredients than most of these, plus sprayed with citrus oil, but it’s essentially soap-based. I have used this product successfully.
************************************************************************

Zum Clean Laundry Soap
64-oz Sweet Orange on Amazon
http://www.indigowild.com/category/natural-home-cleaning/
Liquid laundry soap, various fragrances. No unscented.
************************************************************************

Other Vermont Soap products work for laundry
Vermont Soap Liquid Sunshine Nontoxic Cleaner Concentrate
Gallon Vermont Soap Liquid Sunshine on Amazon
http://shop.vermontsoap.com/Liquid-Sunshine-Nontoxic-Cleaner-Concentrate_p_59.html#.UZBrGo7DIVQ
Vermont Soap has other liquid products that can be used in the laundry. Their website describes uses for each product.
*
************************************************************************
Dri-Pak Soap Flakes
through MSO Distributing
- Dri-pak soap flakes on Amazon
- http://www.msodistributing.com/shop/orders.html
Pure soap flakes may work better if combined with baking soda, washing soda, and borax, per many recipes on the web
************************************************************************
*

Dr. Bronner’s
32-oz Dr. Bronner’s Baby Mild liquid on Amazon
For many people, the only easy soap to access is Dr. Bronners. Unscented Baby Mild as probably the best place to start. Combine with baking soda, borax, and washing soda for better efficacy in laundry.
************************************************************************

Traditional French Soap
Savon de Marseille Soap Flakes
Marius Fabre Marseille soap flakes on Amazon
Traditional soap flakes. Can be combined with washing soda and borax, especially in hard water. (It may pay to shop around, these are expensive. Their liquid soaps use minimal ingredients and pure olive oil, and are also an option combined with washing soda and borax.)
************************************************************************
Pure Soapworks Liquid Laundry Soap or Laundry Powder (Canada)
http://www.puresoapworks.com/specialtysoaps.htm
This company seems to carry a full line of soap products.
************************************************************************
Zote Laundry Soap Flakes
Pack of 8 (17.63oz) Zote laundry soap flakes on Amazon
Still the most popular laundry product in Mexico. (From animal tallow. Also contains fragrance and optical brighteners. Despite it being soap, users must take care if allergy to fragrance or optical brighteners might be an issue.)
************************************************************************
Grandma’s Laundry Soap

Grandma’s laundry soap on Amazon
http://grandmaspureandnatural.com/laundry
A lard-based soap flake product, so, not vegetarian. This appears not to have any dyes or unnecessary ingredients. This company also sells a soap-based stain remover stick and bar soaps. They seem to understand the difference between soap and detergent per the site.
************************************************************************
This is a simple mix of the wonderful Ma’Bella soap with washing soda and borax. Some of my European site users have expressed concerns about borax, so I feel I should mention that.