The current contract with top Hollywood studios is set to expire June 30.
Actor Ned Vaughn said while the board dismissed the petition Saturday, he and his fellow petitioners would not simply accept the decision.
"I don't think there is a will to address this issue seriously under the current leadership," Vaughn told the Times. "We're disappointed, but that hardly means we will go away."
Currently, any actor or extra who appears in a SAG production is eligible for membership in the union and allowed to vote on any contract.