LIVERMORE, Calif. — For seven years, Deepika Jalakam sat at home. Bored, unfulfilled and dependent on her husband for every dime, she struggled with the notion that her professional life was doomed in the land of opportunity.

So when the employment card arrived in the mail in 2015, Ms. Jalakam did what she often does when good fortune comes her way: She placed it before the gods in the Hindu shrine mounted in her kitchen cabinet, blessed it with a dab of red “kum kum” powder and recited a prayer of gratitude.

Within weeks, Ms. Jalakam, who has a degree in biotechnology, landed a job as an analyst at an insurance company. The next year, she and her husband, Vinay Kumar, a software engineer, bought a house. In 2017, the finances of the Indian immigrant couple were secure enough that they decided to have a second child.

All that planning, though, is in jeopardy. Ms. Jalakam and thousands of other spouses of skilled workers have been told that their special work permits — authorization that can mean the difference between struggling and thriving in their adopted homeland — are likely to be revoked.