You can sometimes fill out a welfare application online, depending on where you live and what benefits you are requesting. Other times you can print out applications that need to be handed into a welfare office in person.
The best way to find out whether or not you can fill out an online application for welfare benefits is to visit your state's health and human services department. Some places will allow you to fill out the application online, and some will not. For example, New York City's welfare system requires that you print out and hand in all applications for welfare benefits. Pennsylvania, on the other hand, allows you to apply for benefits online through its COMPASS program.
The term "welfare" in the United States typically refers to Temporary Assistance to Needy Families, a system that gives cash to (usually single parent) families. However, welfare also refers to health care, social security and numerous other benefits.