Some counties (for example the county I work in) provide at home personal care aides and nursing through Medicaid. Institutionalization is usually covered by most forms of Medicaid.
Assuming this person has children, if it’s a woman who is the paralyzed person, the government would first look to the father of the children to take care of them. If the father were deceased or unknown, the government would first look to family and friends (and I am aware that the person in question does not have family, so they would look for friends who might know the child and would want to take the child). If the paralyzed person were the father, they would look to the mother and her family to take the children.
People often assume the government is looking to take children into foster care immediately. Foster care is the last option, and not an easy option regardless of circumstances. An easier (less labor intensive) option is always to place the child through formal custody (i.e. Family Court) with family or friends.