Illegal immigrants can’t vote. Permanent residents can’t vote either. You can only vote once you acquire citizenship (which means doing the whole immigration process, then waiting several more years to be able to apply for citizenship).
My wife was brought here when she was eight years old, and was able to apply for a quasi-legal status when Obama enacted DACA. When we got married, we were frustrated to find out that we couldn't actually apply for her green card because she was still considered "illegal".
Eventually we got an attorney and she was able to find a waiver for us to apply for a green card, but that took three years of waiting. She finally is qualified to apply for citizenship now, and she has, but we're waiting (probably another five to six months) for an update on that.
When we got married, everyone acted like I was some sort of idiot because "obviously" you get a green card automatically if you marry a citizen. Apparently people are getting their understanding of immigration law from sitcoms.