In a working-class suburb of the Egyptian capital, Kamal sells suitcases and handbags out of a small shop. He also delivers U.S. dollars stashed in plastic bags once a rate is agreed to over the phone.

The 45-year-old dealer is part of a black market that is flourishing in the North African country as traditional sources of hard currency—investment and tourism—dry up. Despite attempts by authorities to clamp down, Egypt’s black market provides a critical spigot for dollars and euros to buy such essentials as medicines and...