The President can avoid a shutfown.
Article I of the US Constitution does not say that Congress has the duty to pass appropriations.
Article II declares that the President shall “faithfully execute the Office of President” and “preserve” the Constitution (Section 1 oath) and that “he shall take Care that the Laws be faithfully executed (Section 3). These duties are unconditional.
If the Congress does not appropriate money for the executive branch, it will then cease to function and recede into limbo—unless we interpret the constitution as mandating the functioning of the two branches, in which case to preserve the constitution, the President should keep the executive branch fully functioning, including paying all employees, paying all debts, and contracting with vendors, etc.