Polling is taking place in a Kuwaiti council by-election in which women are allowed to vote for the first time.
Dr Khader calls it a proud day for Kuwait's women
Two women are also among eight candidates running for the seat in the Salmiya district, south of the capital.
The 28,000 eligible voters, 60% of whom are women, are voting in segregated polling booths, a condition demanded by Islamist and tribal MPs.
Women were granted equal political rights last year and will vote in full legislative polls in 2007.
Voting was reported to have begun slowly, as Tuesday is a normal working day in the conservative, oil-rich state.
Kuwait's first women candidates are 32-year-old Jenan Boushehri, a chemical engineer at the Kuwait Municipality, and 48-year-old Khalida Khader, a US-educated physician and a mother of eight.
Bahrain: Constitutional monarch, universal suffrage, political parties banned
Kuwait: Constitutional emir, first elected parliament
Oman: Absolute monarch, elections to consultative bodies
Qatar: Constitutional emir, first to allow women's vote in municipal election
Saudi Arabia: Absolute monarch, consultative elections, but no women's vote
United Arab Emirates: Federation of unelected sheikhs
"I am so pleased that I have become one of the first Kuwaiti women candidates to run in elections," Dr Khader said in an interview with AFP news agency.
"I have broken the ice and hope this will benefit the cause of women."
Women voters quoted by news agencies reflected the years of frustration which this election finally dispels.
"They have given us some attention. We became equal," said voter Iman al-Issa talking to AP.
"It's certainly a historical moment for me. I felt very happy while casting my vote," Afaf Abdullah told AFP outside a polling station.
"I had participated in co-operative society elections before, but the feeling here is totally different. I feel that justice has been achieved for Kuwaiti women."
Jenan Boushehri is one of two female candidates
Despite the segregated voting, women were required to show their faces to judges supervising the elections for the purposes of identification.
There are reports of at least one woman refusing to remove her Islamic veil and leaving the polling station without voting.
The Salmiya seat of the Kuwait Municipal Council fell vacant when incumbent Abdullah al-Muhailbi was named a minister in the Kuwaiti cabinet formed in February.
Attempts by the ruling Sabah family to change the male-dominated legislative structure succeeded in May 2005 - after being blocked for six years by tribal and Islamist members of the National Assembly.
Prime Minister Sheikh Nasser al-Muhammad al-Sabah said on Tuesday that women suffrage boosts Kuwait's international standing.
"We say to our Kuwait sisters, 'Forward, and take your place with your Kuwaiti brothers'," he said in a statement.