Board President Neil Campbell told the audience that the board would "recess this meeting until a later date." He did not say when the meeting would resume.
The board has four days to fill the Region I seat, which represents Coatesville and Valley Township.
According to the Pennsylvania School Code, the board must fill the seat with a majority vote within 30 days of a vacancy. If the board does not make a decision, Chester County Court will fill the seat.
Taylor was the first board member to learn that the district's superintendent, Richard Como, and athletic director Jim Donato had exchanged racist and sexist text messages about students and staff, a revelation that led both men to resign early in the school year.
District residents accused Taylor and the rest of the board of not acting quickly enough when they learned of the texts.
Taylor received 11 percent of the Region I vote in November. Beckershoff drew 34 percent, and the winner, Deborah L. Thompson, received 55 percent.
Thompson said she had not known before the meeting that four of her colleagues would abstain. "I think they want the status quo," she said.
Thompson, Kimberly Mammel, Diane Brownfield, and Stuart Deets voted to appoint Beckershoff. Campbell, Vice President James Fox, Laurie Knecht, and Richard Ritter abstained.
After the meeting, Fox said he needed more time to decide whom to vote for. He said he had not had a vote in mind when the meeting started but decided to abstain after hearing from residents and Thompson.
Several district residents asked the board to follow voters' wishes.
Greg Axe, a write-in candidate who lost in November, said that voters showed they did not want Taylor on the board.
"To place her back in this position undermines everything our community has done in the request for change," he said. "Do what is right for once. Listen to what the community tells you they want by placing Bob Beckershoff in the open seat."