The Constitution vest all executive authority on the president. The president can delegate that authority. That's what all is happening here. Within the executive branch the president has practically total power, hardly if at all possible to constrain by statute, and that's by design in the Constitution.

The president needs the Senate's "advice and consent" to hire principal officers, and does not need the Senate's "advice and consent" for certain other officers as specified by statute. The US Digital Service ("DOGE") is an agency where he did not need the Senate's advice and consent.

The president does NOT need the Senate's advice and consent to fire anyone in the executive branch. For principal officers this was established by the failed impeachment of Andrew Johnson for firing a confirmed cabinet secretary nominated by Lincoln. For other officers this was established by judicial precedent fairly recently when Biden terminated two Trump appointees to minor offices and they sued (and lost).

Similarly the president needs the Senate's advice and consent to enter into treaties. The Constitution is silent as to terminating Senate-confirmed executive officers, officers whose appointments did not require Senate confirmation, or treaties (abrogation). It's essentially settled law that the president does not require the Senate's advice and consent for any of those kinds of terminations.

Therefore, under the Constitution and the political and binding judicial precedents, there can be no law "onto which government workers can cling to refuse these requests."