Yes, non-citizens can and do illegally vote, because no one is checking their eligibility.
Mexican national Rosa Maria Ortega was just convicted in Tarrant County, Texas on two counts of felony voter fraud after illegally registering and voting multiple times. She cast five illegal ballots in Dallas County from 2004 through 2014 (a three-year statute of limitations prevented prosecution for the three earlier offenses).
Ortega was able to commit these state (and federal) felonies by simply checking a box on a form that said she is a citizen. No one checks whether the check boxes are true, no one verifies if applicants are in fact eligible citizens before signing them up to vote. It’s an honor system.
Ortega was only caught when she moved from Dallas to Tarrant County and illegally registered there by again checking the box claiming to be a citizen – after first submitting an application (which was rejected) acknowledging that she’s not a citizen. Tarrant County election officials were required by law to accept Ortega’s second application but contacted the district attorney’s office to initiate an investigation.
Ortega was sentenced to eight years in prison and a $5,000 fine, and after serving her time will be subject to deportation. Strong penalties won’t act as a deterrent, though, if ineligible voters know they’re unlikely to be caught.
Until the voter registration honor system is changed, non-citizens can and will continue to vote.