There has been a spate of reports of rape in West Bengal in recent months, some of which led to public protests against the government for inaction and apathy over women's safety in the state.
Avaaz said it wanted to launch a billboard campaign in Kolkata aimed at provoking Chief Minister Mamta Banerjee to address the issue.
The poster the group wanted to display features a picture of the chief minister with the words "Mamta Banerjee: Stop attacking protesters. Start attacking the problem. Stop rape in West Bengal."
But the five leading outdoor advertising firms refused to run the campaign on the grounds the government would persecute them if they agreed to display the posters on their billboards, said Avaaz.
"There is a regime of self-censorship prevailing in West Bengal now," Avaaz's Alaphia Zoyab said, adding that one state-owned advertiser also declined to run the posters.
"All the outdoor advertising firms pleaded helplessness to run the campaign, which was aimed at drawing the attention of the chief minister on her daily route between her home and workplace in the city."
Avaaz said the advertisements were designed to express the concerns of the more than 17,500 people who signed a petition about the spate of rapes in West Bengal. The campaign did not violate any laws, it added.
Levels of gender violence and domestic abuse are high in patriarchal India and cases of rape and sexual crimes are on the rise.
Reports of rape, dowry deaths, molestation, sexual harassment, kidnappings, trafficking and other crimes against women in the country rose 6.4 percent in 2012 from the previous year. The police logged reports of 244,270 crimes against women in 2012 compared with 228,650 in 2011, according to the National Crimes Records Bureau (NCRB).
West Bengal recorded the highest number of gender crimes at 30,942 - 12.7 percent of the total recorded crimes against women.
One of the top Kolkata-based advertising firms confirmed it had declined to promote the advert.
"We did not respond to their request. We immediately stopped thinking of offering our billboards," said a senior official at the firm who asked to remain anonymous, adding his company was worried about the government's reaction.
But a spokesman for West Bengal's ruling Trinamool Congress party, dismissed the allegations, saying the media and charities were trying to tarnish the government's reputation.
"A few New Delhi NGOs (non-governmental organisations) are up to these smear campaigns," said Trinamool Congress lawmaker Derek O'Brien, without referring to any particular charity.
Avaaz's Zoyab, however, said the advertising campaign is a direct response to Banerjee's handling of the June gang rape and murder of a young village student about 20 kilometres from Kolkata.
After the incident, Banerjee lashed out at protesters, telling them to "shut up", branding them "Maoists" and arresting and detaining 13 activists, said Zoyab.