Federal authorities are struggling to crack down on what they describe as a widespread scheme that has already likely defrauded the Internal Revenue Service of billions of dollars using the stolen identities of Puerto Rican citizens.

The perpetrators of the scheme, authorities say, swipe the Social Security numbers of Puerto Rican citizens, who don't have to pay federal income tax—and are less likely to be on the IRS radar—and use their information to file fake returns. In some cases, they enlist U.S. mail carriers to intercept...