Minecraft wiki entry for Axolotl_farming, associated text: A chest full of shulker boxes helps keep your inventory of buckets manageable.
A conduit allows you to breathe and see better underwater for the purpose of hunting tropical fish and breeding axolotls.
Without a conduit, a helmet enchanted with Respiration is a must when hunting for two axolotls to start breeding. You can obtain these via zombie farming in Hard difficulty, or you can use an anvil to combine a helmet with an enchanted book obtained via fishing or trading.