Congress gave the president broad authority to regulate the immigration process. Trump's executive order cites two statutes cited below.

I'm not a lawyer so it's possible, even likely that there's something I'm missing but to my laymans reading of the law it would seem to me he has the authority to put basically whatever process he wants into place.

Section 215(a) of the Immigration and Nationality Act (INA), codified at 8 U.S.C. § 1185(a), states exactly:

Unless otherwise ordered by the President, it shall be unlawful—(1) for any alien to depart from or enter or attempt to depart from or enter the United States except under such reasonable rules, regulations, and orders, and subject to such limitations and exceptions as the President may prescribe;

Section 212(f) of the Immigration and Nationality Act (INA), codified at 8 U.S.C. § 1182(f), states exactly:

Whenever the President finds that the entry of any aliens or of any class of aliens into the United States would be detrimental to the interests of the United States, he may by proclamation, and for such period as he shall deem necessary, suspend the entry of all aliens or any class of aliens as immigrants or nonimmigrants, or impose on the entry of aliens any restrictions he may deem to be appropriate.