Alligators are vertebrates, not invertebrates. They belong to the phylum Chordata and the subphylum Vertebrata—animals with a backbone—and are reptiles in the order Crocodilia (family Alligatoridae).
The U.S. Integrated Taxonomic Information System (ITIS) places the genus Alligator under Subphylum Vertebrata and Class Reptilia, the same hierarchy used for the American alligator (Alligator mississippiensis) and the Chinese alligator (Alligator sinensis). Invertebrates lack that vertebral column, so alligators do not fit that group.
Source: ITIS taxonomic report for genus Alligator (TSN 174366)