Backers wanted more clarity before the board passed the proposal, which would inform the district's employees how to respond if they were contacted by immigration officials.
But board members first debated whether matters were urgent enough to pass the resolution Tuesday.
The push for the district to support undocumented students comes after school districts in other states have passed similar resolutions.
Unidad Inmigrante members worked with board members and Superintendent Alexa Cunningham on the resolution's language.
But two members of the group opened the meeting by asking the board to avoid a vote until the resolution covers which actions employees could take to deter immigration officials' interference with students' education.
"We would feel much more comfortable having these things addressed in one go, so as to show the community that this is a wholesome, legitimate commitment [with] legal backing," Dominguez said after the meeting.
The board's president, Heather Bennett, said such resolutions were rare. The group typically drafts letters stating the board's position, she said.
She said she was surprised the resolution's supporters wanted members to wait for a vote. President Donald Trump's administration has taken steps to prioritize deporting undocumented immigrants, and the resolution's supporters said community members were uneasy.
"[This] is a really important thing to move forward with," Bennett said.
The initial draft that supporters proposed "had several things in it that we felt we couldn't include, not because we didn't wish we could offer those protections, but because we're not legally able to do that," Bennett said.
Bennett wasn't the only one who said she feared inaction.
"I also worry about delaying in a way that doesn't reflect the urgency," board member Melissa Ford said.
Edison Elementary School Principal Laurie Lacy confirmed through a district spokesman Tuesday night that the people who had guns and Homeland Security vests were in the school's parking lot for about 20 minutes.
Tuesday's resolution would have instructed employees to immediately contact the district's executive director of policy and legal services if contacted by immigration officials or state or local officials acting on behalf of the federal government. The executive director wouldn't grant agents access to students "unless required to do so by law."