The application was sought by opposition parties and a number of civic organisations who believe a bid to remove the president via a motion of no confidence can only succeed if members of parliament (MPs) are allowed to vote in secret.
For a vote of no confidence to succeed, between 50 and 70 ANC MPs would need to vote alongside the opposition parties.
The UDM sought the initial application in the Constitutional Court which has since been backed by the EFF, IFP and the Institute for Security Studies.
Parliament and Zuma have indicated they would oppose the application.
Zuma's midnight cabinet cull in March during which he fired Finance Minister Pravin Gordhan, among others, saw the rand take a nosedive and the country's investment status downgraded to "junk" by two of the world's biggest ratings agencies.
The applicants in the Constitutional Court case have until May 8 to submit supporting documentation, and written submissions by the applicants and the respondents may be filed until May 12.
The hearing is set to begin at 10am on May 15.