No... executive orders are not laws, they can only command the federal government, not individuals or corporations. Meaning this is mostly pointless unless you're using models hosted by the government.
The judicidial branch, so the courts. The government would have to sue the corporation to try to get them to do something, at which point (hopefully) the judge would strike it down.
What courts? Look at all that's been happening over the past months. How much of it have the courts been able to meaningfully impact, vs what's still in effect?
Executive orders aren’t laws (an important fact that should be repeated often and loudly). However, there’s probably room for the executive branch of the government to influence model hosts, as a major funder and consumer.
You left out the part containing the “barrels of money” incentive.