Once you're invited to their house, you can touch the furniture and then talk to the villager repeteadly until they say something close to: "Hey, you like my (item)? I could part with it for (whatever ammount of bells) if you want it." After that, they should let you buy some items from their house.I didn't realize this. How can you get them to do it? I want every single thing in Shino's house so I'd love for her to offer to let me buy things.
I don't know if it's related, but my newer villagers (one moved in yesterday and the other moved in today) haven't invited me into their houses yet.