Staffing agencies have a variety of business models - they may function as the employer of the employee or serve as a matchmaker for other employers who hire employees.
Staffing agencies must create E-Verify cases for all their new hires. This means that if an E-Verify-enrolled staffing agency is required to complete a Form I-9 when hiring, the staffing agency is also required to create an E-Verify case for that new hire. Staffing agencies cannot avoid creating an E-Verify case for an employee merely because a client requests that the staffing agency not create an E-Verify case.
If the relationship is such that the staffing agency is not the hiring employer (i.e., not required to complete a Form I-9), then the staffing agency is not responsible for E-Verify case creation for that individual.
You may wish to contact your legal counsel for help in determining whether your relationship with the individual requires you to complete Form I-9 and create an E-Verify case.