All mammals make distress calls such as when the young are separated from their mothers, and all land animals are technically capable of producing tears to lubricate their eyes. That's not necessarily crying, however.
Charles Darwin claimed that animals such as the Indian elephant or macaque monkey could cry emotional tears, and some pet owners claim their dogs can cry in this way. Additionally, in 2014 an elephant named Raju who was badly abused for 50 years and kept on a chain along with spikes was photographed crying when he was rescued. When he was transferred to an animal sanctuary out of harm's way, let's just say that his "eyes leaked water." I call that crying!
Modern scientists, however, claim that humans are the ONLY animals that cry as a form of emotional expression.