The National Labor Relations Board recently said organizers had proved they had enough support to request an election, but the dates and terms of the vote hadn’t been determined.

A spokesman for Amazon said the company has remained skeptical that there is sufficient worker support to warrant an election, but that the company has “always said that we want our employees to have their voices heard, and we hope and expect this process allows for that.”

The union didn’t immediately respond to a request for comment. Albany organizers have said they want to “send a loud message” to Amazon about exercising their rights to organize for improved working conditions.

Union supporters at the Amazon facility named ALB1 said they want to raise wages and improve safety measures at the facility. The warehouse employs roughly 400 people, according to the National Labor Relations Board.

The Amazon Labor Union has pushed to unionize Amazon facilities throughout the U.S. after the union’s successful bid at the Staten Island warehouse, an effort that established the first union inside Amazon’s U.S. operations. The group has had mixed results since then. It lost a union election in May at a second Staten Island facility, while other organizing efforts haven’t taken off yet.

The union has also been immersed in a legal battle over the results of the first Staten Island vote, which Amazon contested. An official with the National Labor Relations Board who heard arguments from the parties during a weekslong federal hearing recently said the election results should stand.

Organizers at ALB1 face a tough task. In the other elections, Amazon made workers attend presentations to try to dissuade them from voting in favor of the union, employees said. Many workers also do their jobs in isolation, making it more difficult to organize, according to employees.

The union says it has a chance to succeed because one of the leaders of the effort, Heather Goodall, is an employee at ALB1. ALU organizers were successful in their first election in Staten Island because some lead organizers were also employed at the facility, giving them credibility among workers, employees said. Their bid at the second facility failed in part because the union didn’t have strong enough ties with those workers, employees said.

Write to Sebastian Herrera at Sebastian.Herrera@wsj.com