It depends on the console you are playing on.

Nintendo Switch: Yes, you can access and publish custom levels from the underground levelNET computer even if you don't have a Nintendo Online subscription. You just won't be able to play with others in online lobbies.

PlayStation: No, you cannot access or publish custom levels from the underground levelNET computer if you don't have a PS Plus subscription.

Xbox: No, you cannot access or publish custom levels from the underground levelNET computer if you don't have a Xbox Live subscription.