That would make the executive orders illegal, or at best invalid. It wouldn't make DOGE illegal. DOGE is private citizens giving their opinions about how the government should work to a politician who happens to be listening to them (for now, but probably not long given that politician's track record for getting into feuds with former allies.)

Citizens voicing their opinions about the government is clear cut First Ammendment activity, of precisely the sort the first ammendment was intended to protect in the first place. People need to get a grip.