Born in Wilmington, Delaware, she made a name for herself at a young age as the founder of that city's first public kindergarten and for her efforts to introduce child labor laws in that state. In 1883, she founded an organization, now known as the West End Neighborhood House that originally provided social services to Wilmington's immigrant Irish and German families. Nevertheless, she avoided politics and was closely identified with the anti-suffragist movement. In 1900, she testified before the United States Senate Committee on Woman's Suffrage, arguing that women had no place in politics.
Several years later, in 1907, she was drawn to the cause of helping people with tuberculosis (TB). She had already heard of an idea in Denmark in which people attached a special stamp to their mail, the proceeds of which would go to fight the disease, and decided to introduce the same idea in Delaware. Her goal was to raise $300 for a local sanitarium, using a bright red stamp she designed herself, and convinced local post offices to sell them for just 1 cent. This way, she believed, even the poorest people could help in the fight against TB.
Though the idea failed at first, Bissell was able to gain enough publicity from a Philadelphia newspaper to make $3,000, ten times the amount she originally hoped to get. People were intrigued by the idea of Christmas Seals, and the following year, Howard Pyle, a notable illustrator from Wilmington, donated the design of the second stamp.
Bissell spent the remainder of her life promoting Christmas stamps and helping to eliminate tuberculosis. She died in 1948. A public hospital outside Wilmington bears her name.
In 1980, on the 119th anniversary of her birth, the U.S. Postal Service issued a 15 cent stamp in her honor.