No. Arizona tried passing a law that, among other things: criminalized failure to comply with federal alien registration requirements, criminalized working without being authorized to work in the United States, and authorized state officers to arrest aliens without a warrant if they had probable cause that the alien had committed a crime that made them deportable. All three provisions were struck down in Arizona v. United States.
The federal government has "occupied the field" on most immigration issues. That means they've regulated it so extensively that there is zero room for states to act independently. One of Arizona's laws that was struck down exactly duplicated a federal criminal statute, but even that went too far by allowing the state to apply its own enforcement priorities and prosecute cases the federal government would not. If a state made it a crime to be unlawfully present (which is not a federal crime), that intrudes even further on the federal immigration scheme.
This doesn't mean a state can't alert the federal government to people who are unlawfully present. It doesn't necessarily mean state officers can't arrest for federal immigration crimes: a previous Ninth Circuit decision held that Arizona officers could arrest for federal immigration crimes on the same basis that they could arrest for state crimes, and the Supreme Court in Arizona v. US explicitly didn't address the question. However, if state officers make an arrest for a federal crime, the federal government still gets to decide whether or not to prosecute. What you're asking about would remove that federal control, so it is preempted by federal law.