No, legal alien residents are not allowed to vote. My female cousin, who never married (marrying an American when she was younger would have been the easy way to acquire citizenship), has been here 40+ years, is now retired, owns a home, but has never bothered to apply for citizenship, she gets SSAN retirement and a small pension from her former career. Applying for citizenship used to be free of charge, now I hear it is somewhere like $1,000+ and she can’t afford to apply. She cannot vote.
She can advocate, campaign, and donate to her favorite causes and candidates.