The tonic wine - linked to violent crime in Scotland - is made at Buckfast Abbey in Devon.
Campaign group the National Secular Society say the alcoholic beverage is harmful and an “abuse of the charitable system.”
The Buckfast Abbey Trust does not pay tax on its income as it is a charity and made a record £8.8million last year from sales.
The drink has been made by the Benedictine monks of Buckfast Abbey in Devon since the 1920s when the Benedictine brothers developed tonic wine.
In 2011, a new winery was completed at the abbey.
In an interview with BBC News, its vice president Alistair McBay said: “The monks should be setting an example as a religious organisation but the opposite is happening.
“The question needs to be asked ‘Are they serving God or Mamon?”
The trust justifies its existence as a charity in its annual report, stating its aim is the “advancement of the Roman Catholic religion”.
In a statement, it said it would contact the Charity Commission.
A Scottish sheriff said last year there was a “very definite association between Buckfast and violence”.
The Charity Commission said it took “all complaints about registered charities seriously” and would “assess the information about the Buckfast Abbey Trust to determine if there is a regulatory role for the Commission”.