In its recommendation, ISS pointed to criticism of Moderna and Pfizer's "alleged prioritisation of profits" over global public health, and significant support for a waiver of patent protection for Covid-19 vaccines at the World Trade Organization.
But ISS acknowledged that the proposal to commission a report would not force the companies to transfer the technology.
Glass Lewis also recommended investors vote for the proposal at Moderna. But it sided with the board at Pfizer, suggesting a third-party report might not give shareholders more insight than the company's own disclosures.
ISS and Glass Lewis also recommended that shareholders refuse to ratify Moderna's auditors EY, raising concerns that the Boston-based biotech has spent too much on fees unrelated to the audit, which could cause a potential conflict of interest.
ISS said shareholders should not support the re-election of incumbent board members including Moderna chief executive Stéphane Bancel because of concerns over the auditor. Glass Lewis disagreed, noting it was only the first year that Moderna's non-audit fees were higher than the audit fees.
Moderna on Tuesday said that it maximised its manufacturing capacity, adding there is now an "ample supply" of vaccines globally and the problem is distributing these jabs.
The company also said it is "confident" that its auditor is independent, explaining that the non-audit fees were related to the company's rapid expansion last year and will be lower in the future.
Pfizer and EY did not respond to a request for comment.
Pfizer has previously recommended shareholders vote against the proposal, saying it has rapidly expanded production and will send 1bn doses to developing countries this year.
The increased pressure on Moderna comes as it unveiled the first clinical data on a vaccine that combines the original strain of Sars-Cov-2 with the Beta variant.
The new vaccine elicited more than twice the number of neutralising antibodies against Omicron than the original jab, with Bancel saying the data validated the "bivalent strategy" that Moderna has been working on since February 2022, showing it could induce higher antibody levels.
But he said the company was pursuing a vaccine that combined the original strain with the Omicron variant as its lead candidate for a booster in the northern hemisphere autumn. It expects data on that combination in the second quarter.