Zuma was sitting on parliament's board recommendations, "while the term of the interim board that did its best to turn around the ailing public broadcaster has expired," the party said in a statement.
The names of 12 recommended board members were sent to Zuma by the National Assembly just over three weeks ago.
But Zuma, according to a report by the Sunday Times at the weekend, has been reluctant to sign off on the names put forward as he does not trust some of the candidates.
Zuma's spokesman, Bongani Ngqulunga, said the president was "still applying his mind" and a due diligence process was under way to ensure the candidates were fit and proper.
The state broadcaster reported a net loss of just under R1 billion this year.
"President Zuma's inaction has opened a governance vacuum at the SABC and must be viewed by all South Africans as a cause for serious concern. The SACP calls on the President to appoint the SABC board as recommended by Parliament as a matter of urgency and avoid wasting any further time," the SACP said.
The presidency and the SABC earlier this week denied claims by the Democratic Alliance (DA) that staffers at the broadcaster had been "summoned" to the presidency. The meeting was described as a "routine engagement with journalists to discuss operational issues and working relations."
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