The nursing bra as we know it today was not invented by one person specifically, but rather it has evolved over time through the combined efforts of different people and companies. The first nursing bras were likely simple adaptations of regular bras, with the addition of a flap or opening in the cup to allow for easy breastfeeding. Check out this catalog listing for a nursing corset for $0.89!
During the 20th century, maternity and nursing bras started to become more specialized and mass-produced, and companies began to market them as separate products. Some of the first companies to produce nursing bras include Sears Roebuck, and various department stores that started to carry specialized products for nursing mothers. As technology and materials have evolved, nursing bras have become more comfortable and functional with the addition of stretchy fabrics, adjustable straps, and one-handed clasps.
It's hard to point to one inventor of the nursing bra, but it's clear that the nursing bra has been the result of the evolution of technology, materials and the needs of breastfeeding mothers.