I'm assuming they're trying to find an entrance because they're trying to find the headquarters for organization XIII. This is why they're scouring Twilight Town, and eventually the Old Mansion; so they can find an entrance to TWTNW. Hope that helped!
Sora wants in to find Riku and Kairi. He pretty much begs Saix to let him in with this explanation. Mickey probably wants to find TWTNW to find Ansem the Wise and Riku, plus shut down the Organization so they can stop messing with the other worlds.