This one is so simple it requires advanced degrees in both Law and English to misunderstand.
The duty of the President is to enforce lawfully passed, Constitutionally authorized, acts of Congress. The Constitution gives to Congress the specific and SOLE authorization to makes laws regarding immigration and naturalization. Therefore the President MUST enforce those laws, else be guilty of dereliction of duty. [See note at end for exception.]
Dereliction of duty is a crime and misdemeanor ("high" being the status of the office rather than a descriptor of the type of crime or misdemeanor) which requires impeachment and, if guilty, punishment up to removal from office.
Executive orders are allowed only to permit a President to carry out his Constitutional duties. It is not a method of circumventing the legislative branch or any other aspect of the Constitution. Any executive order side-stepping proper Constitutional methods of governance is a usurpation of powers not granted, and are actionable by impeachment of such a President.
Note: The only time this would not apply would be if Congress should pass (and a then-sitting President should sign) a law regarding immigration that violates the Constitution in some other fashion. To use a silly example, say Congress passed a law stating that all immigrants from Poland be awarded free kielbasa for life. Congress has no authority to conscript the goods of others (and is prohibited from doing so by various sections of the Constitution) therefore the Constitutional duty of the President would require such a law not to be enforced.