Thinking of catching a Wobbuffet to use on my team, but how what do you absolutely HAVE to do before you can enter?
The Dark Cave can be accessed as early as Route 31, just before Violet City. However, to truly explore, you need four TMs and the entry from Blackthorn City.
Hope I helped!
You can enter Wobbuffet's room as soon as you defeat Morty and get a Pokemon that knows surf. The room is northeast from Violet City, but you can access only a small part of it before you reach Blackthorn City.