Shampoo as we know it was not introduced until the 1930s. Earlier shampoo consisted of soap, often shaved and melted with the addition of herbs and essential oils. Products offered as shampoo often came in powder from in the early 1900s. In those days people did not wash their hair as frequently as they do today. In 1908, a New York Times article claimed that washing the hair every few weeks was a safe and newly popular practice.
I can't give you exact dates for brands, but several were produced around that time including:
Wella in 1880
Canthrox in 1914
Harmony was sold by Rexall Drug Stores in 1914
C. L. Hamilton produced Cocoanut Shampoo in 1909
Mulsified Cocoanut Oil for Shampooing was available in 1918
Hans Schwartzkopf offered a powdered shampoo in Germany between 1914 - 1920. In 1927 he offered a liquid shampoo
Lustre Creme was available in the 1920s
You can learn more about the history of hair care products in "The Encyclopedia of Hair: A Cultural History" by Victoria Sherrow or in "The American Beauty Industry Encyclopedia" by Julie A. Willett.