I'm an E-Verify employer agent with several clients that were just awarded federal contracts that contain the FAR E-Verify clause. Will I be required to take a tutorial every time I update a clients' status?

No. You and your general users are only required to take the federal contractor tutorial once after you update the first client to Federal Contactor with FAR E-Verify clause.  As is the case with other users, you may also later be required to take additional training after system upgrades or for other reasons.