Open your inventory and you can push a button to switch to the ship. It works anywhere on the ship. If you aren't sure which button opens the ship's cargo (I have it set to Q but I think that's a custom keybinding) it will say at the bottom of the inventory screen which button to press to open the cargo hold.
You can also access the ship's cargo when interacting with merchants and sell things directly from the ship.
Nintendo is now saying that they aren't planning to do another game in the BotW/TotK style, saying that they consider TotK to be the final form of "that version of Zelda"