The use of a relatively obscure visa to enter the U.S. by the Pakistani woman involved in last week’s terror attack in California has raised new concerns about the country’s visa programs.

Tashfeen Malik and her husband, Syed Rizwan Farook, killed 14 people and wounded 21 others in San Bernardino, Calif., in the deadliest terror-related mass shooting on U.S. soil since the Sept. 11, 2001, attacks.

Ms....