They can; some students may feign illness to avoid school, for example. But in cases when a child is involved in malingering, a parent or parents are typically pulling the strings. There have been multiple documented cases of parents going to great lengths to convince others that a child is seriously ill—shaving their head to make them appear to be undergoing cancer treatment, claiming they need treatment for nonexistent seizures, or giving them sleeping pills to make them lethargic—with the specific criminal goal of fundraising for their “care” or applying for unjustified medical benefits.
Some parents may make a child sick to draw attention to themselves, a form of child abuse sometimes referred to as “Munchausen by proxy,” and described in the DSM-5 as factitious disorder imposed on another, but when the motivation is financial and not emotional, malingering is involved.