Contact the Social Security Administration in person. The child will be eligible only if "totally and permanently disabled," so be prepared to gather all the medical records.
You can apply in person, at your local Social Security office, or you can apply online. Here is a link from the Social Security Administrations website explaining to you how to apply:...
SSI recipients are not liable for current, ongoing child support and child support, current or past-due, cannot be withheld from SSI payments. [SSI recipients are, of course, liable for any past due...