In early April, Payal Raj accompanied her family to India to renew the visas that permit them to live in the United States. She and her husband waited until they had been vaccinated, carefully preparing their paperwork according to the advice of their immigration lawyers. But the visa itself would soon strand her in India indefinitely, separating her from her husband and daughter in Hendersonville, Tenn. “Our family is in a crisis,” said Ms. Raj, who is one of thousands of immigrants stuck in India, in part because the Biden administration’s restrictions on most travel from the country mean that temporary visa holders are explicitly barred from re-entering the United States. “Every morning is a struggle.” The restrictions, issued as a devastating surge in coronavirus cases has overwhelmed India in recent weeks, prohibit Ms. Raj and others like her from returning to their homes, families and jobs in the United States. Even those exempt under the ban are in limbo as the outbreak forces the U.S. Embassy and consulates to close, leaving many with no clear path home.