Is it illegal to live in the United States without Social Security and if so how can you report someone?
No, you are not required to participate in the Social Security System.That said, most, but not all employers, require their employees to participate in the system, and therefore if you choose to work for that particular company you need to apply for a Social Security Number and use it in their employment. (Some city governments, such as my city, do not have to require all employees to participate in Social Security which is why I, on my 23rd year as a police officer, have never paid into Social Security and will never receive it.)Another class of people that choose not to participate in Social Security are some self-employed business people--if you own the company and you're the only employee you do not need to require yourself to participate in Social Security.
Does every citizen have to have a Social Security card? No, but if they are going to be claimed as a dependent on their parent's income tax, or file their own income taxes, or collect most types of disability, or attend most schools, then they will need one. It's almost unfathomable that an adult US citizen would not have one.