Tech entrepreneur Cris Mercado still keeps the envelope that he stuffed in late 2014 with school records, his proof of residence and other documents he gathered to apply for a new program to shield him from deportation and enable him to work legally in the U.S.

“Whatever it took I wanted to be the first to apply,” recalled the 35-year-old undocumented Filipino, who arrived in New York when he was 5.

Mr. Mercado’s hopes were put on hold when a federal judge last year blocked the implementation of President Barack Obama’s new policy after 26 states filed suit challenging it.

On Monday, the Supreme Court is scheduled to hear oral arguments on the policy in one of its biggest cases this term, though the February death of conservative Justice Antonin Scalia leaves the court at risk of ties. If it deadlocks 4-4, the lower-court injunctions remain in place, likely freezing the policy for the remainder of the Obama administration.

The policy is called the Deferred Action for Parents of Americans and Lawful Permanent Residents, or DAPA. It would offer temporary protection from deportation and authorize work permits to about five million undocumented immigrants, most of them parents of U.S. citizens.