They want to pressure the CNE, which they say is stalling their attempts to hold a recall referendum to oust President Nicolas Maduro.
The CNE had been expected to announce on Tuesday whether the first step in the referendum process had been met.
Instead, it said it would meet on 1 August for further discussions.
Venezuela is going through a dire economic crisis and recent polls suggest 64% of Venezuelans would vote to remove Mr Maduro from office.
So far, the opposition has completed the first step. On 2 May, it handed the CNE a petition with almost two million signatures, many more than the 194,729 needed at this first stage.
The CNE has since been working on step two, validating the signatures by asking signatories to come forward and be fingerprinted.
On Tuesday, the CNE had been expected to announce the final result of the validation and to set a date for step three, when signatures will have to be collected on a second petition, which, if successful, would trigger the referendum proper.
But by the end of Tuesday, the CNE released a statement (in Spanish) saying it would only meet on 1 August to examine the auditors’ report on the validation process.
It also warned the opposition that it would not be pressured and that it “would suspend its activities in the case of any disturbances to the public order”.
Despite the warning, opposition leader Henrique Capriles repeated his call for his supporters to join a march to the CNE headquarters in Caracas on Wednesday and for those outside the capital to protest in front of their local CNE offices.
Timing is key
The opposition has long accused the CNE of siding with the government and using delaying tactics to try and thwart the referendum.
Opposition leaders are anxious to proceed with the referendum as soon as possible because its timing is key to what happens next.
Should it be held before 10 January and go against Mr Maduro, fresh elections will be triggered.
But if the vote were to be held after 10 January – in the last two years of Mr Maduro’s mandate – he would be replaced by his vice-president and supporter, Aristobulo Isturiz.
The referendum could also be delayed by legal challenges launched by government supporters.
On Tuesday, government officials asked the electoral authorities to suspend the opposition coalition behind the recall referendum for alleged fraud.
They said that thousands of names on the recall petition belonged to dead people.
It is not clear at this point what effect a possible suspension of the opposition coalition would have on the drive for the recall referendum.