In order to establish the statutory excuse and avoid a civil penalty if you are found to be employing an illegal worker, the right to work check must be carried out by the employer.
This means that you cannot rely on a third party performing the check for you, such as a recruitment agency.
For British and Irish citizens with a valid passport (or Irish passport card), you can engage an Identity Service Provider to carry out digital identity checks but you, as the employer, still remain liable for the civil penalty should the employee later be found to be working without the requisite permission and it is reasonably apparent the check has not been completed correctly.