The Clicks advert, which has since been withdrawn, was posted online last week.
"[I am] deeply disappointed that we allowed insensitive and offensive images to be published on our website," Clicks boss Vikesh Ramsunder said in a statement on Sunday.
He added that the "negligent employees" had been suspended.
His apology has not assuaged EFF supporters who raided several stores across the country on Monday.
Videos shared on Twitter show some Clicks stores closed, while others have security guards forming a protective line in front of singing protesters.
In another video the party supporters are seen pulling down shelves in a store.
In Mpumalanga province, a group of people threw a petrol-bomb at a Clicks store in Emalahleni municipality.
"According to information that we have gathered, around 1am a group of men came and threw a petrol bomb at the shop. However, only a few glass [panes] were cracked but the shop itself is safe, nothing was damaged in the shop," said the province's police spokesperson Leonard Hlathi.