Authorized representative is a different hat
Nothing in federal law requires a notary to complete an I-9, and nothing in a notary commission authorizes it. Employers hire notaries because they are locatable, screened, and used to examining identification, not because a notarial act is involved. The practical consequence is that the usual notary habits have to be switched off: no seal, no journal entry, no notarial wording, and no fee charged against the state's notarial fee schedule.
A few states complicate the picture. Non-attorney assistance with immigration paperwork is regulated in states including California, and California notaries are separately barred from advertising as a notario público because the term implies legal authority abroad. Check your state's rules on immigration-document services before accepting recurring I-9 work, and never let the engagement drift into advising the employee about their status.
