CAPE TOWN, South Africa—MasterCard Worldwide this week started issuing credit cards in Somalia in a bid to tap the $1.6 billion annual remittances business that has been disrupted by U.S. and U.K. banks suspending their services on fears of inadvertently being involved in money laundering or terrorist financing.

MasterCard is linking up with Somalia’s Premier Bank and has already distributed its first credit cards to Somalis, after a three-year process to set up shop in the strife-torn nation in the Horn of Africa, said Daniel...