I don't think that is correct. Since it would be an EO and not a law, it only applies to Executive branch agencies. He would simply direct those agencies to not do business with any company that does not report.
It would be up to the company to take it to court as a restraint of trade issue I would guess.
You’re absolutely right; my mistake. I guess that the issue would get to court if a company sued the agency for discrimination or restraint of trade or whatnot, or if someone at an agency was reprimanded or fired for continuing to do business with a company that wasn’t complying with the EO.