Political groups want all four remaining election commissioners to quit.
About 50 people belonging to various political groups Monday marched to parliament and submitted the demand to National Legislative Assembly (NLA) president Pornpetch Witchitcholchai.
Besides the resignations, the political activists also wanted the scrapping of the new electoral system, calling it an undemocratic method chosen by "a handful of people".
Meanwhile, Mr Pornpetch, a member of the committee responsible for selecting five of the seven members of the Election Commission (EC), said the panel might have to approach people to apply for posts if candidates do not come forward and submit applications.
The NLA needs to pick new election commissioners after military-appointed legislators rejected all seven candidates selected to serve on the commission in a vote on Feb 22.
In the event applicants are deemed unqualified, the selection panel is likely to tap suitable people, said Mr Pornpetch, adding the new selection process must be a unanimous decision. The committee will consider the matter after the application process ends on April 8, he said.
It was widely reported that the NLA had received "a clear signal" from the regime to turn down all seven candidates. Critics claimed that the Supreme Court selection process was not open and could have faced a legal challenge.
Under the new constitution, the Supreme Court picks two of the seven EC members and the remaining five are chosen by a panel.
Supreme Court Chief Justice Chatchai Janpraisri, and Pakorn Mahannop, a judge on the Supreme Court, were chosen as the two candidates under the court's quota, in a closed-door meeting.
Former EC member Somchai Srisutthiyakorn said the Supreme Court's choice of Mr Chatchai and Mr Pakorn may have violated Section 12 of the organic bill on the composition of the EC. Section 12 requires the court to hold the vote openly.
However, the NLA recently amended the organic law governing the EC to allow candidates, chosen by the Supreme Court but rejected by the NLA, to re-apply with the Supreme Court.
Mr Pornpetch said Monday he did not know the reason behind the NLA's decision. "We have to look at the minutes of the NLA meeting to find out," Mr Pornpetch said.