No, that is impossible without a new law. They only have authority over federal employees.
You do not understand “law” and the powers of a president.
The President is the Chief Executive Officer of the Executive Branch of Government.
His job - and his sworn Constitutional duty - is to “execute” (or “enforce”) the laws passed by Congress.
Congress passed a law that requires ALL employees in the USA to fill out an I-9 Work Authorization Form.
Congress empowered the Department of Homeland Security to review and verify EVERY Form I-9 in the country.
If you don’t understand that, I have no idea what else to say to you.