The members voted 137 to 30 in favour of the resolution in a special meeting held in Fredericton.
The meeting was organized under the society's rules after it received a petition with more than 200 names opposing a decision in June by its governing council to recognize future graduates of Trinity Western University.
The Christian school requires all students and staff to sign a covenant barring same-sex relationships.
Law society president Helene Beaulieu said the result of Saturday's vote will be taken to the next council meeting on Sept. 26 to discuss the implications of the resolution.
"Council values the opinion of all the membership," said Beaulieu. "I am confident that council will work through this difficult and controversial issue with openness and transparency."
She said if the resolution is adopted, students from the university will not be eligible for admission to the bar in New Brunswick.
However, it is unclear whether the outcome of the vote is binding on the law society, which has said it will need a legal opinion.
Earl Phillips, Trinity Western's executive director, expressed concern with the members' resolution.
"Difficult decisions involving fundamental rights and freedoms should not be decided by popular vote," said Phillips. "There is no evidence to suggest that the religious beliefs that guide TWU would affect the ability of its law graduates to serve all clients."
To date, bar associations in Alberta and Saskatchewan have approved accreditation — although Saskatchewan has put its decision on hold along with Manitoba.
Law societies in Ontario and Nova Scotia voted against accreditation, which caused the school to challenge those decisions in the courts in both provinces.
Separate judicial reviews will be held in December in the Ontario Superior Court and in the Nova Scotia Supreme Court.
The school's accreditation as a teaching institution by the B.C. government is also the subject of a lawsuit by a group of lawyers in the school's home province.
The law school is scheduled to open in the fall of 2016.