http://www.colorincolorado.org/article/your-rights-parent-pu... makes it seem like there is a US law which prohibits a school from asking for an ID. ("The people who work at the school are not allowed to ask parents to show an ID or to give a social security number. You don't have to answer questions about your immigration status or your child's.")
http://www.nj.com/cumberland/index.ssf/2014/04/aclu_takes_so... says that ID isn't required to enroll a student into the school district. More specifically:
> In the letter to various school districts, Shalom cites the 1982 United States Supreme Court case Plyler v. Doe, which decided that undocumented children living in this country cannot be precluded from public elementary and secondary schools because of immigration status.
> ... “We were surprised by the number and broad range of school districts with restrictive, discriminatory enrollment policies,” said Jeanne LoCicero, deputy legal director of the state ACLU. “These policies exist in almost every county, from the poorest cities to the wealthiest suburbs, and everywhere in between. Clearly, the scale of this problem demands serious action, and serious self-examination among New Jersey school districts.”
You can ask what their policy is for undocumented people or those without id.
If they ask "aren't you a US citizen?" you can say "you are not legally required to know that."
However, while I suspect that may let you pick up your child, meet the teacher, and attend PTA meetings, I don't know if you might not be allowed to chaperone or volunteer.
From one of Skenazy's posts (who schoen mentioned), that may require background checks, as in http://www.freerangekids.com/no-mam-you-may-not-chaperone-yo... .