There are a wide range of frogs, salamanders and caecilians that can be kept as pets in an aquarium. Many of these are not found in the pet trade. This is usually because they're either too big for most commercial aquariums (ex: giant salamanders), are endangered (ex: achoques), haven't entered the pet trade yet, or a mix of the three.
All of the animals on this list are fully aquatic, meaning they can live without access to land. However, many still need to breathe air, as they lack gills in their adult phase.