Maria Zamudio grew up in the shadow of S.B. 1070, the divisive Arizona immigration law passed by Republicans in 2010 that sparked years of fear and protests. Friends fled the state, fearing the immigration enforcement provisions that made it known as the “show me your papers law.” She worried her undocumented parents would be deported while she was at school.

“It took away our childhood,” said Ms. Zamudio, 24. She joined thousands of young Latinos who jumped into politics to fight the law and celebrated when it was hobbled by legal challenges. “I thought we were over this.”

But now, Republicans in Arizona believe that widespread discontent over President Biden’s performance on immigration has given them a new opening to confront unauthorized immigration — unease that would put the border crisis directly onto the ballot in November.

Republican lawmakers are pushing for a ballot measure that would make unlawfully entering Arizona from Mexico a state crime. The legislators’ proposal would give local police officers the power to arrest and jail migrants and would allow state judges to deport them. It would also make officers and other government officials immune from any resulting lawsuits.

“We have to do something,” State Senator Sonny Borrelli, the Republican majority leader, said in a fiery floor speech before a vote in May advancing the plan. “We have an invasion.”