The unusual call, from organizations including Oxfam and Global Witness, comes the week before the board of the Extractive Industries Transparency Initiative is due to hold a periodic meeting in Oslo.
Launched in 2002 by then British Prime Minister Tony Blair, the initiative compels oil and gas companies to disclose the payments they make to governments around the world in leases, taxes and royalty payments -- with the aim of deterring theft, embezzlement, bribery and other malfeasance.
Corruption in oil, gas and mineral-rich nations frequently starves public education, health care and infrastructure of the wealth produced by extractive industries.
But as part of its regulatory rollback, President Donald Trump's administration announced in November it would exit EITI, claiming that enforcing the standard was not consistent with US law.
The decision meant the United States joined Azerbaijan and Equatorial Guinea in refusing to participate. Fifty-one countries currently implement EITI.
Exxon and Chevron have refused to disclose their US tax payments, the authors of Wednesday's letter said, calling this "a repeated and willful violation" of the initiative's code of conduct and an "act of bad faith."
The companies have also lobbied against a US securities regulation that likewise compelled publicly traded oil companies from disclosing payments to governments, according to the letter, which was addressed to former Swedish Prime Minister Frederik Reinfeldt, the current chair of the EITI board.
Trump also scrapped the regulation last year.
"We recommend strongly that these companies be removed from the EITI board for violating the EITI code of conduct, as well as the spirit of EITI," said the letter, which was also signed by representatives of EG Justice, Publish What You Pay and the Project on Government Oversight.
In addition to Exxon Mobil and Chevron, representatives of BP, Royal Dutch Shell, Rio Tinto, Total and Statoil also sit on the EITI board.